TWO IMPORTANT HYMNS OF THE VEDAS.

The Vedas are full of philosophical inquiry and I am giving two very important suktas from the Rigveda. There is no idol worship in the Vedas and Gods of nature like Indra, Agni, Varuna, Ushas etc are worshiped. Yet they make a distinction between gods and creation and the supreme being who they name as Brahman. The Brahman of the Vedas is unknowable and is beyond our intellect. 

NASADIYA SUKTA.

Also known as Hymn of Creation in English and is in the Rigveda. It is concerned with cosmology and the origin of the Universe.

This is hymn full of philosophical inquiry into how the creation came into being. It says that even the supreme lord may not know how the creation has arisen.  

The hymn goes

“In the beginning not even nothingness existed nor existence. What covered it (the thing which we can never say whether it existed or did not exist) and then where was it”.  

“There was neither death nor immortality then and nor the heavens”.

“The one breathed windlessly and there was nothing else but that one”.

“But, after all, who knows, and who can say whence it all came, and how creation happened?”

“The gods themselves are later than creation, so who knows truly whence it has arisen?”

“Whence all creation had its origin, he(the supreme), whether he fashioned it or whether he did not”.

He, who surveys it all from the highest heaven, probably he knows - or maybe even he knows not.

 PURUSHA SOOKTA.

The Purusha sukta is from the Rigveda and gives a description of the spiritual unity of the universe. It also points to the birth of the varna system.

It presents the nature of Purusha or the cosmic being as both inherent in the manifested world and yet transcendent to it.

From this being, the sukta holds, the original creative will proceeds which causes the projection of the universe in space and time.

The Purusha sukta, in the seventh verse, hints at the the four classes of the society.

THEORY OF RELATIVITY IN SIMPLE TERMS-II

This is the second note in the series of 3 notes made for my daughter to explain the Theory of Relativity. 

Imagine that 2 trains were standing side by side in a station. If our train starts, we feel that the other train is moving. If the other train moves, we feel that our train is moving. If you really want to know which train is moving, then you have to look to the other side say at the station or a telegraph pole to find out which train is moving.

If two trains cross each other, then to the observer in the train it appears that the other train is passing very rapidly. If two trains are traveling in the same direction at the same speed, then both the trains appear stationery unless we look to the other side.

If we are traveling in a train with all the shutters closed and if we are traveling in a straight-line, and the train has perfect shock absorbers and does not accelerate or decelerate, then we would not be able to figure out whether the train is moving or not. Whatever scientific experiments we may conduct on the train we would not be able to find out whether the train is moving or not.

This indicates that movement is relative. There is no absolute motion in the Universe. If you want to find out the velocity of one object, then you have to compulsorily compare it with another object. Suppose there is nothing else in the Universe except the train in which we are traveling, then we cannot even figure out whether the train is moving or not. In such an instance, the difference between movement and inertia ceases.

Relativity of motion is known even before Einstein to Newton as well.  

Earth is like the train we mentioned. If we take the earth alone, then no experiment would be able to tell us whether the earth is moving or not. Only by comparing the position of the earth with other bodies like the Sun, Moon and the stars, we can find the motion of the earth.   

The earth is rotating on its axis at a speed of 1600 Km per hour and it is also revolving around the Sun at a speed of 32 Km per sec. Not just the earth alone, the Sun, the planets and the stars are moving. The solar system is moving compared to our star cluster. The star cluster is moving compared to the Milky Way galaxy. The Milky Way galaxy is moving compared to the other galaxies. This means there is nothing in the Universe that is stationery and every object is moving compared to another. All these are relative motions.

Does that mean, there is no absolute motion in the Universe? Newton felt there is. He assumed that the space surrounding us is not moving and at absolute rest. He felt that by comparing the speeds of the objects with space one can find out the absolute speeds at which they are moving. There is no proof for this concept, yet all the scientists after Newton took it to be true. Not only that, they even postulated that absolute space had certain characteristics and the space is permeated with a substance called Ether.  

Subsequently scientists discovered that light traveled in waves. They then believed a medium is required to transmit the waves and Ether is the medium that transmits light. This concept has brought in many inconsistencies which the scientists were unable to answer and a doubt arose as to whether Ether existed or not.

This doubt was taken up as a challenge by an American scientist named Michelson. There is no other experimental scientist in the nineteenth century who is better than Michelson.  He developed an instrument called interferometer in the year 1887 and along with Edward William Morley he conducted an experiment to establish the presence of Ether. This is known as the Michelson Morley experiment.

If a ship is traveling on a calm sea, the sea recedes back as fast as the ship is traveling forward. Similarly when a ship called earth is traveling in a sea called Ether, Ether has to recede as fast as the earth is traveling forward.

When a ship is traveling we can see the flag on the ship fluttering even when there is no breeze when stationery. If we stand on top of the ship we can feel the wind blowing against us.  

Similarly when the ship called earth is speeding through the sea called Ether, then we have to feel the flowing of Ether. But scientists believed that Ether is so minute that it can even pass through atoms and therefore we do not feel its presence but they assumed that it will certainly bring about changes in the speed of light waves.

In the case of sound waves, their intensity decreases in upwind and increases in downwind. Similarly light waves would have to travel faster in the direction of Ether breeze and slower against the Ether breeze.

Michelson and Morley experimented to find out as to much the speed of light decreases when it is going against Ether and how much it decreases when it is going with the Ether. Scientists knew that light travels approximately at a speed of 300,000 KM per second (298,051 KM/Sec to be exact).  They also knew that earth is revolving around the sun with a speed of 32 Km/Sec.

Therefore if a light wave is sent in the direction of the earth’s revolution around the sun, then its speed has to increase by 32 Km/Sec and if the light wave is sent against the earth’s revolution its speed has to decrease by 32 Km/Sec.  To find this out Michelson used the interferometer which had the capacity to measure changes in the speed of light with an accuracy of 0.4 Km/sec.

In the experiment Michelson sent the light ray against the earth’s revolution, in the direction of the earth’s revolution and also to all other sides but found that the speed of light did not change an iota irrespective of the direction of the wave. With this experiment all the scientists in the world were shocked and assumed that some error has crept into their experiment. Many scientists therefore repeated the experiment but again with the same result.

Now, what are they to assume? Ether existed or not? They believed that it existed for 200 years and could not easily give up the concept and it put them in a quandary and confused them.  

In 1905 Einstein unraveled this mystery when he published his special theory of relativity. He said that Ether does not exist. He also said in his theory that the speed of light is a Universal constant and irrespective of what speed an object is moving and a ray of light emanating from that object is always the same at 300,000 Km/Sec. Nothing in the Universe as per his theory can surpass the speed of light. There is no absolute motion in the Universe and all motion is only relative. There is no absolute space either. Because there are objects we perceive the distance between them is space. If there are no objects, then there is no space either.   

THEORY OF RELATIVITY IN SIMPLE TERMS-I

My daughter asked me about the Theory of Relativity which can be easily understood because she felt confused by the abundance of matter she found on the net about the theory...I wrote this small intro for her with the help of a wonderful Telugu book I have on the subject.......I wrote two more notes to her explaining it a bit more which I would post later sometime......

The Universe is finite but boundless. This sentence appears paradoxical and contradictory, but if we investigate further on the meaning, then the mystery reveals itself.

For example, take the earth. The surface of the earth is finite, yet it is boundless. One may walk or ride miles and miles on the earth but the earth never ends and there is no board saying that this is the end of the earth. But does that mean earth has infinite dimensions? No but it is a sphere and curves in on itself and closes so that we can never reach its end by going on it. 

Einstein says that the Universe is also like this and curves in and closes on itself. This means that the Universe is not infinite but yet it is boundless. But one caution here, one can compare the surface of the earth with that of the Universe but not its volume.

Einstein postulated that like the earth, the Universe curves on itself and closes. The Universe we see is of 3 dimensions (as per our perception) Length, breadth and height. Now Einstein postulates that our Universe has not 3 but for dimensions and the fourth is time.

What is time? It is duration between 2 events. The same way as we measure length in meters and Kilometers, similarly we are measuring time in seconds, minutes and hours. Therefore time is also a dimension. Without specifying the time an event becomes meaningless. For example “Buddha is born”. But when was he born? Until the time frame is supplied it does not become specific to us and complete.

Because of this, time and space cannot be separated and the both have a relationship. This concept was used in science by Einstein. He named it “Space time continuum”. Therefore for Einstein, Universe consists of both space and time and they are interlinked. This discovery was done by Einstein with the Theory of Relativity. As per him there is nothing absolute in the Universe and everything is relative.

For example we can give many examples for relativity in life. Can you call a person only a Father? He is a father only to his children, but to another he is a husband, to another he is a son and to yet another he is a friend.

Big and small are also relative. A mouse is big to an insect but small to a cat. Here we can remember the riddle of the line where a bigger line drawn before it makes it small.  

Similarly left and right are also relative. In a street, if you enter from one side if it is left, then if you enter from another side it would be right. Newton’s theory of gravitation was ruling by Einstein’s time but inconsistencies in the theory started emerging slowly and scientists found them very hard to reconcile.

Then in 1905 Einstein published his special theory of relativity and his General Theory of Relativity in 1915 which satisfactorily answered those inconsistencies that could not be reconciled by Newton’s theory. Einstein’s theory is full of mathematical deductions and it is said that initially very few scientists in the world could understand it. Einstein never entered a laboratory and brought out all his deductions purely with the help of mathematics.  

Einstein is like a prophet. After Einstein postulated, scientists have carried out various tests that proved him to be correct. His mathematical deductions are beyond our comprehension but his theory can be revealed in everyday terms. It seems a student asked his professor as to what is the theory of relativity. The professor said “ If you lover is with you, then an hour seems a second to you, and if she is not there then one second seems like one hour”. So time is but relative.

When Einstein died, he was working on a “Unified Field Theory” that proposed to combine the Theory of Relativity with Electromagnetism. One wonders what mysteries he would have unraveled if he stayed alive. Einstein died in 1955 and unfortunately to this day the Unified Field Theory is incomplete.  

THE PROBLEM IN PALESTINE.

Perhaps, it is time that the Jews living in Palestine and the Palestinians to bury their differences and arrive at a solution. How to go about it is an extremely difficult proposition but the solution has to come from the people living Palestine themselves rather than any third country. Although Jews are settlers in Palestine, they have to be accepted by the Palestinians, and the Jews have to respect palestinian self governance. Intervention of 3rd countries in such conflicts is futile.  

But what was the cause of the problem? Although I knew about it in shreds I wanted to know more about it. I searched the net and found an interesting article on it. This is an abridged version of the article. I retained most of the wording from the article.  

The fight for a Jewish homeland is rooted in 19th Century movements promoting “Zionism”: the forced emigration of Central European Jews – notably from Russia, Poland, Ukraine and Germany – into Palestine, as well as North and South America.

The undisputed founding father of International Zionism was Viennese lawyer Theodor Herzl who convened the First Zionist Congress in Basel, Switzerland in 1898. His seminal book, Ein Judenstaat (“A Jewish State”) published in 1896 sets out the rationale, method and plan for founding a sovereign Jewish State, discretely leaving the door open for founding not just one but two Jewish states.

Cosmopolitan Jews combined their leaders’ immense political, financial, media and diplomatic clout to ensure European Jews would, against all odds, get their homeland in Palestine. This however, resulted in ignoring the interests and lives of millions of Palestinians living there for many generations.

Historically, Christian Europe discriminated against the Jews. For centuries they were second-grade citizens and were systematically expelled from France, Germany, Italy, Spain, Scandinavia, Britain, Portugal, Austria, Hungary, Russia, Belorussia, Poland, Ukraine and many other nations.

Scorned by both Catholics and Protestants, they were segregated inside ghettos from where they focused on retailing and, with time, became Europe’s foremost international bankers.

Resistance against them regularly flared, often into bloody pogroms. Zionist activists took advantage of this state of affairs to rally their flock into their movement. This centuries-long targeting of Jews is called “Anti-Semitism”, and can only be explained in two basic manners:

Though “Anti-Semitism” certainly has its religious overtones, its root cause in modern times is not so much religious as it is social. Contrary to Islam and Christianity which many times rose up in Holy War and Crusading to convert the heathen, Judaism never seeks to convert anybody by force.

Quite the contrary: Jews are such by birth; by blood; by genetics. Religious conversion is therefore not an option.

For centuries, Jews would part company with the hope-filled greeting “Next year in Jerusalem”. Gathering in Zion was a leitmotiv of their hope to one day achieving nationhood in the land of Moses, Abraham and David.

Theodor Herzl, Leon Pinsker, Chaim Weizmann, David Ben Gurion and other founding fathers fought to create Israel however at the turn of the 20th Century; the Holy Land was just not an option. When Zionists petitioned Ottoman Turkey’s Sultan who reigned over Palestine to give them their state he simply refused. But they continued lobbying, leveraging and promoting their ideas and ideals, whilst through academia and the press they nurtured increasingly liberal social conditions necessary for them to flourish.

By then, extremely powerful European bankers with deeply embedded contacts and into every government, directly and indirectly consolidated and promoted the Zionist ideal. They used long-term rather than short-term strategies; their names are symbols of the burgeoning international finance over-world: Rothschild, Warburg, Schiff, Lazard, Bleichroeder, Belmont, Hirsch, Montefiore, Goldschmidt, Oppenheimer, Goldman, Sachs, Erlanger, Speyer, Mendelssohn and many other powerful European bankers, brokers and traders who extended their power and influence throughout Europe and the Americas.

German billionaire Maurice Hirsch supported Herzl’s plans founding the Jewish Colonization Association that promoted emigration of Eastern European Jews to America, notably Argentina, a country ranking very high on Zionism’s priority list.

In World War I, which Germany, Austria and Ottoman Turkey lost in 1918 to France, Britain and the US, leaving Palestine as a British Mandate.

The year before, UK Baron Walter Rothschild secured from Britain’s Foreign Secretary Sir Arthur James Balfour a “Declaration” whereby Britain “would favor the establishment in Palestine of a national home for the Jewish people, and will use their best endeavors to facilitate the achievement of this object” subject to nothing being “done which may prejudice the civil and religious rights of existing non-Jewish communities in Palestine.” This triggered a new wave of European Jewish emigration to Palestine.

The 30’s and 40’s, however, brought a new wave of persecution under the Third Reich and World War II, generating in its aftermath renewed support for a Jewish homeland. That was finally achieved in 1948, not so much as recognition of Zionists’ right to impose a Jewish State upon Palestine, but due to unrestricted Western support for the Zionist Plan. 

CASTE---ITS GENESIS---ITS (IR)RELEVANCE IN THE PRESENT DAY.

No one knows for sure how the caste system has evolved in India. But it can be estimated as to how it has originated.

Caste in India perhaps had its origins in the occupation people pursued. In ancient times mechanization was very low and it was required that the craft had to be learnt very carefully under a guru for acquisition of a particular skill. This is passed down generation after generation. As mobility was very restricted in the ancient times, people initially tended to take up that particular skill which had an expert in their neighborhood. Slowly that entire neighborhood acquired that skill and handed it down generation to generation.

After some time perhaps due to competition arising out of learning the same skill in another neighborhood marriages were preferably arranged in the same neighborhood so that knowledge of that particular skill did not leak out. Slowly with time, the process became more rigid and the neighborhood which practiced a particular skill got formulated into a caste. As they lived together and practiced the same skill and also intermarried, relations within that caste improved and they felt themselves a part of that whole and caste affiliation became stronger.

Similar communities have evolved in other neighborhoods as well. As these other neighborhoods already had an expertise in that particular trade, inter marrying between these neighborhoods appeared alright because it did not pass on the tricks of the trade to others who are new to the trade. Slowly all the neighborhoods practicing that trade evolved into one caste with rigid rules for marriage.

Of course, the same thing could have happened and caste should have evolved in many other countries too, but surprisingly there is no parallel to this in the world.

Sociologists are not able to understand why caste as it is practiced in India, has not evolved anywhere else in the world.

Perhaps at the time it evolved caste system did have relevance and despite having its setbacks, has helped the society to function in a smooth manner. But, in the present day when everyone is free to learn whatever they please, and choose whatever they please, it becomes an anachronism and also irrational.

Even today inter marriages between castes is taboo and we read about the many suicides committed by lovers due to their parents refusal to the marriage. Of course, there are a few cases of marriages between different castes but they are the exceptions.

Some people argue that people from the same caste would have similar habits and therefore it is correct to marry within the caste. But is that really true? At present it is the economic class that influences behavior of the people, and families from the same economic class tend to have a stronger chance of having similar habits rather than families from the same caste.

Why then do people hang on to a system that is outdated? Unfortunately, in our society tradition is never questioned even by the educated classes. When Raja Ram Mohan Roy set up a college for women at Calcutta, he was furiously attacked by the scholars at that time because they felt that girls should not get educated. We know now what a foolish concept that was, but at that point of time it was perfectly accepted by society unquestioningly.

Similarly, he also fought a bitter war against Sati, widow remarriage and also property inheritance rights to women. In this he is way ahead of his times. One would be horrified now to know that there were some13,000 recorded cases of sati in Bengal province alone in a period of some 10 years during Roy’s time.

It is the will of that great man that made him hang on to his beliefs despite all the persecution meted out to him. Can one man change the society? Well, Rama Mohan Roy’s life is an example although even he would have been helpless had the British not supported his actions. Such is the power of tradition.

It is high time now that the caste should be cast aside by our society. Youngsters should be allowed to marry of their own liking and caste should not be a barrier.

The biggest barrier to breaking caste is the concept of caste based reservation. This has ensured that people organize themselves on the basis of caste and exert pressure of the governments to get more and more concessions. This strengthens caste groups and the benefit of their caste is seen as greater than that of the society as a whole. The reservation system has now ensured that caste differences only deepen day by day as each caste tries to grab more and more. Such a scenario is never good for the country as a whole but which political party has the guts to bell this cat? But, of course which political party is bothered about the country?

What India needs now is a strong reformer like Rama Mohan Roy. One can only hope that such a person would come forward and lead our youngsters to a tomorrow, where the word caste gets confined to the history books. Perhaps this is wishful thinking and a sweet dream but what else can one do?

INDIAN MATHEMATICS

Some Indians feel that Indian mathematicians discovered all the mathematical concepts and the Europeans simply took them from India. On the contrary, some European scholars brush aside Indian mathematics and give the credit to Greece. Neither of these views is fully correct.   

After seeing the works of all the Indian mathematicians down the ages, one can be absolutely certain that India had great mathematicians who were original, and had produced concepts that were discovered in the western world much later. 

It is very difficult to say with certainty if the west has learnt and borrowed anything from India in the development of western mathematics except for the numerals, zero and the concept of infinity. But since contact between the cultures of India and the West started with the Bactrian Greeks, it is likely that through them, the Indian thought on Mathematics was transmitted to the west. it is highly probable that much later through the Arab Caliphates of Cordoba Grenada, different ideas have been exchanged and thoughts would have flown both ways. Europeans have also borrowed concepts from Mesopotamia who in turn may have been influenced by Indian thought and also influenced it. 

But while the Europeans went ahead and developed mathematics further, the Indian mathematics stagnated after the 16^th century till Srinivasa Ramanujan came along. 

It is universally acknowledged today that the concept of Zero, Infinity, the decimal system world uses today as well as the so called Arabic numerals are of Indian origin. The Indian numerals hvae gone to Europe through the Arab caliphates in Spain and thus are known as Arabic numerals. 

I have gone through Wikipedia for learning more on Indian mathematics and abridged its content below mathematician wise.

Baudhayana (8^thCentury BC): He composed the Sulabha Sutras( these are the mathematics required for construction of the sacrificial altars of the Vedas).

Baudhayana’s treatise contains a general statement of the Pythagorean Theorem for the sides of a square and also simple Pythagorean triples.  He also gave a square root for the number 2 which is accurate up to 5 decimal places. It should be noted that Baudhayana predates Pythagoras who lived in the 6^th century BC.

There were two other Sulabha Sutras composed by Manava (7^thcentury BC) and Apastamba (6^th century BC).

Pingala(3^rd Century BC): He is a musical theorist who authored a Sanskrit treatise on prosody. He stumbled upon the Pascal Triangle and the binomial coefficients despite not having knowledge of the binomial theorem itself. His work also contains the basic idea of Fibonacci numbers. Although his entire work did not survive, a 10^th century AD commentary on it by Halayudha exists.    

Katyayana(3^rd Century BC): He wrote Katyayana Sulabha Sutra which presented a lot of geometry including the general Pythagorean Theorem and also a computation of the square root of 2 to 5 decimal places.  

Jain Mathematicians(400 BC to 200 AD): The Jaina mathematicians freed the Indian mathematics from its religious and ritualistic contents. They concentrated a lot on the concept of infinity.  

They devised notations for simple powers (and exponents) of numbers like squares and cubes, which enabled them to define simple algebraic equations. Jaina mathematicians were apparently also the first to use the word Shunya (void in Sanskrit) to refer to zero. 

Surya Siddhanta (400AD): The author of this treatise is unknown.It contains the roots of modern trigonometry. It uses Sine, Cosine, Tangent and the inverse Sine.

Aryabhata I (476-550AD): he wrote the Aryabhatiya which contained Quadratic equations, Trigonometry and the value of Pi up to 4 decimal places.  

Varahamihira (505-587AD): He wrote the Pancha Sidhhanta which made contributions to Trigonometry including Sine and Cosine tables up to 4 decimal places and some formulas relating to them.    

Brahmagupta (597-668 AD): In his astronomical work Brahma Sphuṭa Siddhanta , included two chapters devoted to the field of mathematics. Basic operations (including cube roots, fractions, ratio and proportion, and barter)and Practical mathematics (including mixture, mathematical series, plane figures, stacking bricks, sawing of timber, and piling of grain). 

Brahmagupta was the first to give rules to compute with zero.

Brahmagupta gave the solution of the general linear equations.

He also went on to solve systems of simultaneous indeterminate equations.

Brahmagupta's most famous result in geometry is his formula for cyclic quadrilaterals. Given the lengths of the sides of any cyclic quadrilateral,he gave an approximate and an exact formula for the figure's area.

Bhaskara I  (600–680 AD): He expanded the work of Aryabhata in his books titled Mahabhaskariya, Aryabhatiya-bhashya and Laghu-bhaskariya.

Hecame out with:

Solutions of indeterminate equations.

An approximation of the Sine function.

A formula for calculating the Sine of an acute angle without the use of a table,correct to 2 decimal places.

AryabhataII  (c. 920–1000) wrote a commentary on Shridhara, and an astronomical treatise Maha-Siddhanta. The latter discusses Arithmetic, Algebra and solutions of indeterminate equations.

Shripati Mishra (1019-1066) wrote on permutations and combinations,solution of simultaneous indeterminate linear equation, calculating of planetary longitudes, eclipses and planetary transits.   

Bhaskara II (1114-1185) wrote a number of important treatises.Siddhanta Shiromani, Lilavati Ganita, Bija Ganita, Gola Adhyaya and is one of India’s greatest mathematicians. He has contributed on

 1.    Interest computation.

2.    Arithmetic and Geometric progressions.

3.    Plane Geometry.

4.    Solid Geometry.

5.    Solutions of combinations.

6.    Proof for being divided by zero is infinity.

7.    Proof of the Pythagorean Theorem.

8.    Surds.

9.   Solutions of Quadratic equations, Cubic equations, Quartic equations, equations with more than one unknown.

10.  Conceived differential calculus.

11.  Compute Pi to 5 decimal places.

12.  Calculated the length of the earth’s revolution around the sun.   

The Kerala school of astronomy and mathematics (1300-1600 AD) was founded by Madhava of Sangamagrama in Kerala, South India and included among its members:

1. Parameshvara,

2.Neelakanta Somayaji,

3.Jyeshtadeva,

4.Achyuta Pisharati,

5.Melpathur Narayana Bhattathiri,

6.Achyuta Panikkar.

In attempting to solve astronomical problems, the Kerala school astronomers independently created a number of important mathematical concepts. The most important result being the series expansion for trigonometric functions.

The theorems were stated without proof, but proofs for the series for sine, cosine,and inverse tangent were provided a century later in the work written in Malayalam, by Jyesthadeva.

Their discovery of these three important series expansions of calculus, several centuries before calculus was developed in Europe by Isaac Newton and  Leibniz was an achievement.However, the Kerala School did not invent calculus; they developed neither a theory of differentiation or integration, nor the fundamental theorem of calculus.

Varahamihira (505-587AD): He wrote the Pancha Sidhhanta which made contributions to Trigonometry including Sine and Cosine tables up to 4 decimal places and some formulas relating to them.    

Brahmagupta (597-668 AD): In his astronomical work Brahma Sphuṭa Siddhanta , included two chapters devoted to the field of mathematics. Basic operations (including cube roots, fractions, ratio and proportion, and barter)and Practical mathematics (including mixture, mathematical series, plane figures, stacking bricks, sawing of timber, and piling of grain). 

Brahmagupta was the first to give rules to compute with zero.

Brahmagupta gave the solution of the general linear equations.

He also went on to solve systems of simultaneous indeterminate equations.

Brahmagupta's most famous result in geometry is his formula for cyclic quadrilaterals. Given the lengths of the sides of any cyclic quadrilateral,he gave an approximate and an exact formula for the figure's area.

Bhaskara I  (600–680 AD): He expanded the work of Aryabhata in his books titled Mahabhaskariya, Aryabhatiya-bhashya and Laghu-bhaskariya.

Hecame out with:

Solutions of indeterminate equations.

An approximation of the Sine function.

A formula for calculating the Sine of an acute angle without the use of a table,correct to 2 decimal places.

AryabhataII  (c. 920–1000) wrote a commentary on Shridhara, and an astronomical treatise Maha-Siddhanta. The latter discusses Arithmetic, Algebra and solutions of indeterminate equations.

Shripati Mishra (1019-1066) wrote on permutations and combinations,solution of simultaneous indeterminate linear equation, calculating of planetary longitudes, eclipses and planetary transits.   

Bhaskara II (1114-1185) wrote a number of important treatises.Siddhanta Shiromani, Lilavati Ganita, Bija Ganita, Gola Adhyaya and is one of India’s greatest mathematicians. He has contributed on

1.    Interest computation.

2.    Arithmetic and Geometric progressions.

3.    Plane Geometry.

4.    Solid Geometry.

5.    Solutions of combinations.

6.    Proof for being divided by zero is infinity.

7.    Proof of the Pythagorean Theorem.

8.    Surds.

9.   Solutions of Quadratic equations, Cubic equations, Quartic equations, equations with more than one unknown.

10.  Conceived differential calculus.

11.  Compute Pi to 5 decimal places.

12.  Calculated the length of the earth’s revolution around the sun.   

The Kerala school of astronomy and mathematics (1300-1600 AD) was founded by Madhava of Sangamagrama in Kerala, South India and included among its members:

1. Parameshvara,

2.Neelakanta Somayaji,

3.Jyeshtadeva,

4.Achyuta Pisharati,

5.Melpathur Narayana Bhattathiri,

6.Achyuta Panikkar.

In attempting to solve astronomical problems, the Kerala school astronomers independently created a number of important mathematical concepts. The most important result being the series expansion for trigonometric functions.

The theorems were stated without proof, but proofs for the series for sine, cosine,and inverse tangent were provided a century later in the work written in Malayalam, by Jyesthadeva.

Their discovery of these three important series expansions of calculus, several centuries before calculus was developed in Europe by Isaac Newton and  Leibniz was an achievement.However, the Kerala School did not invent calculus; they developed neither a theory of differentiation or integration, nor the fundamental theorem of calculus.

SANKARACHARYA AND ADVAITA.

As per many of the scholars Sankaracharya lived between 788 and 820 AD for a period of just 32 years. His biography isinterlaced with legend and lore and cannot be historically supported. Whatevermay have been the events of his life, he has championed the cause of Advaita and hisbrilliance has few equals amongst the Indian philosophers.  

Sankara was born at Kaladi in Kerala.According to tradition and lore, his parents had been childless for many yearsand prayed in a Shiva temple at Thrissur for a child.

Shiva later appeared to both the husband and wife in their dreams and asked them to choose between a mediocre son with along life and an extraordinary son who would live for a short time. They chose the latter and when the son was born to them whom they named him Sankara.

His father died when Sankara was very young and his Upanayanam was performed by his mother. As a child Sankara showed remarkable scholarship by mastering the 4 Vedas by the age of 8.   

Sankara was attracted towards sanyasa at theage of 8 itself but his mother gave her consent for that only after much persuasion. Legend says that once when he was bathing in the Poorna river, acrocodile caught hold of his leg and it appeared that it would kill him.Sankara then requested his mother to give him consent for sanyasa at least before his death. His mother consented and the crocodile left Sankara and disappeared.

Sankara, then traveled to North India in search of a Guru. On the banks of river Narmada at Omkareswar (MP) he met Govinda Bhagavatpada. When the latter questioned about Sankara’s identity,Sankara came out with a wonderful verse that brought out the Advaita philosophy. Govinda was impressed by Sankara and accepted as his disciple.        

Theguru instructed Sankara to write a commentary on the Brahma Sutras and propagate theAdvaita philosophy. Sankara travelled to Kashi,where a young man named Sanandana,hailing from Chola territory in South India,became his first disciple.

Accordingto legend, while on his way to the Vishwanath Temple atKasi, an untouchable accompanied by fourdogs came in the way of Sankara. When asked to move aside by Sankara'sdisciples, the untouchable replied: "Do you wish that I move my everlasting Ātman or this body made of flesh?"

Realizingthat the untouchable was none other than god Shiva himself, and his dogs the four Vedas,Sankara prostrated himself before him, composing five shlokas.

At Badari he wrote his famous Bhashyas (commentaries).

Oneof the most famous debates of Sankara was with Maṇḍana Miśra. Sankara sought a debate with Kumārila Bhaṭṭa and met him in Prayag where he had buried himself in a slow burning pyre to repent for sins committed against his guru. Kumārila Bhaṭṭa then asked Sankara to proceed to Mahiṣhmati tomeet Maṇḍana Miśra and debate with him instead.

Maṇḍana Miśra held the view that the life of a householder was far superior to that of a monk. This view was widely shared and respected throughout India at that time.

This is totally against what Sankara believed and therefore it is important for Sankara to debate with Mandana.

After debating for over fifteen days, with Maṇḍana Misra'swife Ubhaya Bhāratī acting as referee, Maṇḍana Misra accepted defeat.

Ubhaya Bhāratī then challenged Sankara to have a debate with her in order to'complete' the victory. She asked him questions related to sexual union between man and woman – a subject in which Sankara had no knowledge, since he was a true celibate and sanyasi.

Sankara asked for a "recess" of 15 days. As per legend, he used the art of "para-kaya pravesa" (the spirit leavingone's own body and entering another's) and exited his own body, which he asked his disciples to look after, and psychically entered the dead body of a king and learnt the art of love from the Kings 2 wives. Later, Sankara entered his own body and regained consciousness and he answered all questions put to him by Ubhaya Bhāratī and defeated her.

After this, Sankara began a tour of conquest for thepropagation of the Advaita philosophy by controverting all philosophies opposedto it. He travelled throughout India, from South India to Kashmir and Nepal, preaching to the local populace and debating philosophy with Hindu, Buddhist and other scholars and monks along the way.

This was no mean feat as in those times India is heavily forested and the roads and forests are fully infested with robbers and bandits.Despite this Sankara traveled fearlessly to the 4 corners of India and founded the 4 peethams at Sringeri, Dwaraka, Badrinath and Puri.

He died at the age of 32, a young age but by that time itself, his brilliance has established him as a great philosopher.

Now, what is thephilosophy of Sankara? I try to put it very briefly below.

Ultimate reality according to Sankara is Atman or Brahman which is pure consciousness devoid of all attributes (nirguna) and all categories of the intellect (nirvisesha).

Brahman associated with its potency Maya appears as the qualified Brahman or the lord who is the creator, preserver and destroyer of this world which is his appearance.

Jiva or the individual self is a subject object complex.

Its subject element is pure consciousness and is called Sakhsin.

Its object element is the internal organ called Antahkarana. The source of this internal organ is Avidya which causes individuality.

In liberation Avidya is destroyed by Jnana and Sakshi is realized as the Brahman which it always is.

Maya or Avidya is not pure illusion. It is not only absence of knowledge but also positive wrong knowledge. It is a cross of the real and the unreal. In fact, it is indescribable. It is neither existent, nor nonexistence nor both.

Sankara emphasizes that from the phenomenal point of view,the world is quite real. It is not an illusion. It is the creation of Ishwara. Jiva is ignorant of the essential unity and takes only diversity as true and wrongly regards himself as the agent and the enjoyer. Avidya conceals the unity and projects names and forms. When Jiva realizes this Avidya, Moksha is attained and the final release is attained after death.