A Brief History of Indian Mathematics

http://www.esamskriti.com/html/essay_index.asp?cat=938&subcat=937&cname=history_of_indian_maths
Prof. Vijaya Kumar Murty

MATHEMATICS has played a significant role in the development of Indian culture for millennia. Mathematical ideas that originated in the Indian subcontinent have had a profound impact on the world. Swami Vivekananda said: ‘you know how many sciences had their origin in India. Mathematics began there. You are even today counting 1, 2, 3, etc. to zero, after Sanskrit figures, and you all know that algebra also originated in India.’

It is also a fitting time to review the contributions of Indian mathematicians from ancient times to the present, as in 2010, India will be hosting the International Congress of Mathematicians. This quadrennial meeting brings together mathematicians from around the world to discuss the most significant developments in the subject over the past four years and to get a sense of where the subject is heading in the next four. The idea of holding such a congress at regular intervals actually started at The Columbian Exhibition in Chicago in 1893. This exhibition had sessions to highlight the advancement of knowledge in different fields. One of these was a session on mathematics. Another, perhaps more familiar to readers of Prabuddha Bharata, was the famous Parliament of Religions in which Swami Vivekananda first made his public appearance in the West.

Following the Chicago meeting, the first International Congress of Mathematicians took place in Zurich in 1897. It was at the next meeting at Paris in 1900 that Hilbert formulated his now famous 23 Problems. Since that time, the congress has been meeting approximately every four years in different cities around the world, and in 2010, the venue will be Hyderabad, India. This is the first time in its more than hundred-year history that the congress will be held in India. This meeting could serve as an impetus and stimulus to mathematical thought in the subcontinent, provided the community is prepared for it. Preparation would largely consist in being aware of the tradition of mathematics in India, from ancient times to modern and in embracing the potential and possibility of developing this tradition to new heights in the coming millennia.

In ancient time, mathematics was mainly used in an auxiliary or applied role. Thus, mathematical methods were used to solve problems in architecture and construction (as in the public works of the Harappan civilization) in astronomy and astrology (as in the words of the Jain mathematicians) and in the construction of Vedic altars (as in the case of the Shulba Sutras of Baudhayana and his successors). By the sixth or fifth century BCE, mathematics was being studied for its own sake, as well as for its applications in other fields of knowledge.

The aim of this article is to give a brief review of a few of the outstanding innovations introduced by Indian mathematics from ancient times to modern. As we shall see, there does not seem to have been a time in Indian history when mathematics was not being developed. Recent work has unearthed many manuscripts, and what were previously regarded as dormant periods in Indian mathematics are now known to have been very active. Even a small study of this subject leaves one with a sense of wonder at the depth and breadth of ancient Indian thought.

The picture is not yet complete, and it seems that there is much work to do in the field of the history of Indian mathematics. The challenges are two-fold. First, there is the task of locating and identifying manuscripts and of translating them into a language that is more familiar to modern scholars. Second, there is the task of interpreting the significance of the work that was done.

Since much of the past work in this area has tended to adopt a Eurocentric perspective and interpretation, it is necessary to take a fresh, objective look. The time is ripe to make a major effort to develop as complete a picture as possible of Indian mathematics. Those who are interested in embarking on such an effort can find much helpful material online.

We may ask what the term ‘Indian means in the context of this discussion. Mostly, it refers to the Indian subcontinent, but for more recent history we include also the diaspora and people whose roots can be traced to the Indian subcontinent, wherever they may be geographically located.

Mathematics in ancient times (3000 to 600 BCE)

The Indus valley civilization is considered to have existed around 3000 BCE. Two of its most famous cities, Harappa and Mohenjo-Daro, provide evidence that construction of buildings followed a standardized measurement which was decimal in nature. Here, we see mathematical ideas developed for the purpose of construction. This civilization had an advanced brick-making technology (having invented the kiln). Bricks were used in the construction of buildings and embankments for flood control.

The study of astronomy is considered to be even older, and there must have been mathematical theories on which it was based. Even in later times, we find that astronomy motivated considerable mathematical development, especially in the field of trigonometry.

Much has been written about the mathematical constructions that are to be found in Vedic literature. In particular, the Shatapatha Brahmana, which is a part of the Shukla Yajur Veda, contains detailed descriptions of the geometric construction of altars for yajnas. Here, the brick-making technology of the Indus valley civilization was put to a new use. As usual there are different interpretations of the dates of Vedic texts, and in the case of this Brahmana, the range is from 1800 to about 800 BCE. Perhaps it is even older.

Supplementary to the Vedas are the Shulba Sutras. These texts are considered to date from 800 to 200 BCE. Four in number, they are named after their authors: Baudhayana (600 BCE), Manava (750 BCE), Apastamba (600 BCE), and Katyayana (200 BCE ). The sutras contain the famous theorem commonly attributed to Pythagoras. Some scholars (such as Seidenberg) feel that this theorem as opposed to the geometric proof that the Greeks, and possibly the Chinese, were aware of.

The Shulba Sutras introduce the concept of irrational numbers, numbers that are not the ratio of two whole numbers. For example, the square root of 2 is one such number. The sutras give a way of approximating the square root of number using rational numbers through a recursive procedure which in modern language would be a ‘series expansion’.

This predates, by far, the European use of Taylor series.

It is interesting that the mathematics of this period seems to have been developed for solving practical geometric problems, especially the construction of religious altars. However, the study of the series expansion for certain functions already hints at the development of an algebraic perspective. In later times, we find a shift towards algebra, with simplification of algebraic formulate and summation of series acting as catalysts for mathematical discovery.

Jain Mathematics (600 BCE to 500 CE)

This is a topic that scholars have started studying only recently. Knowledge of this period of mathematical history is still fragmentary, and it is a fertile area for future scholarly studies. Just as Vedic philosophy and theology stimulated the development of certain aspects of mathematics, so too did the rise of Jainism. Jain cosmology led to ideas of the infinite. This in turn, led to the development of the notion of orders of infinity as a mathematical concept. By orders of infinity, we mean a theory by which one set could be deemed to be ‘more infinite’ than another. In modern language, this corresponds to the notion of cardinality. For a finite set, its cardinality is the number of elements it contains. However, we need a more sophisticated notion to measure the size of an infinite set. In Europe, it was not until Cantors work in the nineteenth century that a proper concept of cardinality was established.

Besides the investigations into infinity, this period saw developments in several other fields such as number theory, geometry, computing, with fractions and combinatorics. In particular, the recursion formula for binomial coefficients and the ‘Pascal’s triangle’ were already known in this period.

As mentioned in the previous section, astronomy had been studied in India since ancient times. This subject is often confused with astrology. Swami Vivekananda has speculated that astrology came to India from the Greeks and that astronomy was borrowed by the Greeks from India. Indirect evidence for this is provided by a text by Yavaneshvara (c. 200 CE) which popularized a Greek astrology text dating back to 120 BCE.

The period 600 CE coincides with the rise and dominance of Buddhism. In the Lalitavistara, a biography of the Buddha which may have been written around the first century CE, there is an incident about Gautama being asked to state the name of large powers of 10 starting with 10. He is able to give names to numbers up to 10 (tallaksana). The very fact that such large numbers had names suggests that the mathematicians of the day were comfortable thinking about very large numbers. It is hard to imagine calculating with such numbers without some form of place value system.

Brahmi Numerals, The place-value system and Zero

No account of Indian mathematics would be complete without a discussion of Indian numerals, the place-value system, and the concept of zero. The numerals that we use even today can be traced to the Brahmi numerals that seem to have made their appearance in 300 BCE. But Brahmi numerals were not part of a place value system. They evolved into the Gupta numerals around 400 CE and subsequently into the Devnagari numerals, which developed slowly between 600 and 1000 CE.

By 600 CE, a place-value decimal system was well in use in India. This means that when a number is written down, each symbol that is used has an absolute value, but also a value relative to its position. For example, the numbers 1 and 5 have a value on their own, but also have a value relative to their position in the number 15. The importance of a place-value system need hardly be emphasized. It would suffice to cite an often-quoted remark by La-place: ‘It is India that gave us the ingenious method of expressing all numbers by means of ten symbols, each symbol receiving a value of position as well as an absolute value; a profound and important idea which appears so simple to us now that we ignore its true merit. But its very simplicity and the great ease which it has lent to computations put our arithmetic in the first rank of useful inventions; and we shall appreciate the grandeur of the achievement the more when we remember that it escaped the genius of Archimedes and Apollonius, two of the greatest men produced by antiquity.

A place-value system of numerals was apparently known in other cultures; for example, the Babylonians used a sexagesimal place-value system as early as 1700 BCE, but the Indian system was the first decimal system. Moreover, until 400 BCE, THE Babylonian system had an inherent ambiguity as there was no symbol for zero. Thus it was not a complete place-value system in the way we think of it today.

The elevation of zero to the same status as other numbers involved difficulties that many brilliant mathematicians struggled with. The main problem was that the rules of arithmetic had to be formulated so as to include zero. While addition, subtraction, and multiplication with zero were mastered, division was a more subtle question. Today, we know that division by zero is not well-defined and so has to be excluded from the rules of arithmetic. But this understanding did not come all at once, and took the combined efforts of many minds. It is interesting to note that it was not until the seventeenth century that zero was being used in Europe, and the path of mathematics from India to Europe is the subject of much historical research.

The Classical Era of Indian Mathematics (500 to 1200 CE )

The most famous names of Indian mathematics belong to what is known as the classical era. This includes Aryabhata I (500 CE) Brahmagupta (700 CE), Bhaskara I (900 CE), Mahavira (900 CE), Aryabhatta II (1000 CE) and Bhaskarachrya or Bhaskara II (1200 CE).

During this period, two centers of mathematical research emerged, one at Kusumapura near Pataliputra and the other at Ujjain. Aryabhata I was the dominant figure at Kusumapura and may even have been the founder of the local school. His fundamental work, the Aryabhatiya, set the agenda for research in mathematics and astronomy in India for many centuries

One of Aryabhata’s discoveries was a method for solving linear equations of the form

ax + by = c. Here a, b, and c are whole numbers, and we seeking values of x and y in whole numbers satisfying the above equation. For example if a = 5, b =2, and c =8 then x =8 and y = -16 is a solution. In fact, there are infinitely many solutions:

x = 8 -2m

y = 5m -16

where m is any whole number, as can easily be verified. Aryabhata devised a general method for solving such equations, and he called it the kuttaka (or pulverizer) method. He called it the pulverizer because it proceeded by a series of steps, each of which required the solution of a similar problem, but with smaller numbers. Thus, a, b, and c were pulverized into smaller numbers.

The Euclidean algorithm, which occurs in the Elements of Euclid, gives a method to compute the greatest common divisor of two numbers by a sequence of reductions to smaller numbers. As far as I am aware Euclid does not suggest that this method can be used to solve linear equations of the above sort. Today, it is known that if the algorithm in Euclid is applied in reverse order then in fact it will yield Aryabhata’s method. Unfortunately the mathematical literature still refers to this as the extended Euclidean algorithm, mainly out of ignorance of Aryabhata’s work.

It should be noted that Aryabhata’s studied the above linear equations because of his interest in astronomy. In modern times, these equations are of interest in computational number theory and are of fundamental importance in cryptography.

Amongst other important contributions of Aryabhata is his approximation of Pie to four decimal places (3.14146). By comparison the Greeks were using the weaker approximation 3.1429. Also of importance is Aryabhata’s work on trigonometry, including his tables of values of the sine function as well as algebraic formulate for computing the sine of multiples of an angle.

The other major centre of mathematical learning during this period was Ujjain, which was home to Varahamihira, Brahmagupta and Bhaskaracharya. The text Brahma-sphuta-siddhanta by Brahmagupta, published in 628 CE, dealt with arithmetic involving zero and negative numbers.

As with Aryabhata, Brahmagupta was an astronomer, and much of his work was motivated by problems that arose in astronomy. He gave the famous formula for a solution to the quadratic equation

It is not clear whether Brahmagupta gave just this solution or both solutions to this equation. Brahmagupta also studied quadratic equation in two variables and sought solutions in whole numbers. Such equations were studied only much later in Europe. We shall discuss this topic in more detail in the next section.

This period closes with Bhaskaracharya (1200 CE). In his fundamental work on arithmetic (titled Lilavati) he refined the kuttaka method of Aryabhata and Brahmagupta. The Lilavati is impressive for its originality and diversity of topics.

Until recently, it was a popularly held view that there was no original Indian mathematics before Bhaskaracharya. However, the above discussion shows that his work was the culmination of a series of distinguished mathematicians who came before him. Also, after Bhaskaracharya, there seems to have been a gap of two hundred years before the next recorded work. Perhaps this is another time period about which more research is needed.

The Solution of Pell’s equation

In Brahmagupta’s work, Pell’s equation had already made an appearance. This is the equation that for a given whole number D, asks for whole numbers x and y satisfying the equation

Xsquare – Dysquare = I.

In modern times, it arises in the study of units of quadratic fields and is a topic in the field of algebraic number theory. If D is a whole square (such as 1, 4, 9 and so on), the equation is easy to solve, as it factors into the product

(x- my ) (x + my) = 1

where D = m square. This implies that each factor is + 1 or – 1 and the values of x and y can be determined from that. However, if D is not a square, then it is not even clear that there is a solution. Moreover, if there is a solution it is not clear how one can determine all solutions. For example consider the case D=2. Here, x = 3 and y=2 gives a solution. But if D=61, then even the smallest solutions are huge.

Brahmagupta discovered a method, which he called samasa, by which; given two solutions of the equation a third solution could be found. That is, he discovered a composition law on the set of solutions. Brahmagupta’s lemma was known one thousand years before it was rediscovered in Europe by Fermat, Legendre, and others.

This method appears now in most standard text books and courses in number theory. The name of the equation is a historical accident. The Swiss mathematician Leonhard Euler mistakenly assumed that the English mathematician John Pell was the first to formulate the equation, and began referring to it by this name.

The work of Bhaskaracharya gives an algorithmic approach ------- which he called the cakrawala (cyclic) method ------ to finding all solutions of this equation. The method depends on computing the continued fraction expansion of the square root of D and using the convergents to give values of x and y. Again, this method can be found in most modern books on number theory, though the contributions of Bhaskaracharya do not seem to be well-known.

Mathematics in South India

We described above the centres at Kusumapara and Ujjain. Both of these cities are in North India. There was also a flourishing tradition of mathematics in South India which we shall discuss in brief in this section.

Mahavira is a mathematician belonging to the ninth century who was most likely from modern day Karnataka. He studied the problem of cubic and quartic equations and solved them for some families of equations. His work had a significant impact on the development of mathematics in South India. His book Ganita– sara– sangraha amplifies the work of Brahmagulpta and provides a very useful reference for the state of mathematics in his day. It is not clear what other works he may have published; further research into the extent of his contributions would probably be very fruitful.

Another notable mathematician of South India was Madhava from Kerala. Madhava belongs to the fourteenth century. He discovered series expansions for some trigonometric functions such as the sine, cosine and arctangent that were not known in Europe until after Newton. In modern terminology, these expansions are the Taylor series of the functions in question.

Madhava gave an approximation to Pie of 3.14159265359, which goes far beyond the four decimal places computed by Aryabhata. Madhava deduced his approximation from an infinite series expansion for Pie by 4 that became known in Europe only several centuries after Madhava (due to the work of Leibniz).

Madhava’s work with series expansions suggests that he either discovered elements of the differential calculus or nearly did so. This is worth further analysis. In a work in 1835, Charles Whish suggested that the Kerala School had laid the foundation for a complete system of fluxions. The theory of fluxions is the name given by Newton to what we today call the differential calculus. On the other hand, some scholars have been very dismissive of the contributions of the Kerala School, claiming that it never progressed beyond a few series expansions. In particular, the theory was not developed into a powerful tool as was done by Newton. We note that it was around 1498 that Vasco da Gama arrived in Kerala and the Portuguese occupation began. Judging by evidence at other sites, it is not likely that the Portuguese were interested in either encouraging or preserving the sciences of the region. No doubt, more research is needed to discover where the truth lies.

Madhava spawned a school of mathematics in Kerala, and among his followers may be noted Nilakantha and Jyesthadeva. It is due to the writings of these mathematicians that we know about the work of Machala, as all of Madhava’s own writings seem to be lost.

Mathematics in the Modern Age

In more recent times there have been many important discoveries made by mathematicians of Indian origin. We shall mention the work of three of them: Srinivasa Ramanujan, Harish-Chandra, and Manjul Bhargava.

Ramanujan (1887- 1920) is perhaps the most famous of modern Indian mathematicians. Though he produced significant and beautiful results in many aspects of number theory, his most lasting discovery may be the arithmetic theory of modular forms. In an important paper published in 1916, he initiated the study of the Pie function. The values of this function are the Fourier coefficients of the unique normalized cusp form of weight 12 for the modular group SL2 (Z). Ramanujan proved some properties of the function and conjectured many more. As a result of his work, the modern arithmetic theory of modular forms, which occupies a central place in number theory and algebraic geometry, was developed by Hecke.

Harish-Chandra (1923- 83) is perhaps the least known Indian mathematician outside of mathematical circles. He began his career as a physicist, working under Dirac. In his thesis, he worked on the representation theory of the group SL2 (C). This work convinced him that he was really a mathematician, and he spent the remainder of his academic life working on the representation theory of semi-simple groups. For most of that period, he was a professor at the Institute for Advanced Study in Princeton, New Jersey. His Collected Papers published in four volumes contain more than 2,000 pages. His style is known as meticulous and thorough and his published work tends to treat the most general case at the very outset. This is in contrast to many other mathematicians, whose published work tends to evolve through special cases. Interestingly, the work of Harish-Chandra formed the basis of Langlands’s theory of automorphic forms, which are a vast generalization of the modular forms considered by Ramanujan.

Manjul Bhargava (b. 1974) discovered a composition law for ternary quadratic forms. In our discussion of Pell’s equation, we indicated that Brahmagupta discovered a composition law for the solutions. Identifying a set of importance and discovering an algebraic structure such as a composition law is an important theme in mathematics. Karl Gauss, one of the greatest mathematicians of all time, showed that binary quadratic forms, that is, functions of the form

axsquare + bxy + cysquare

where a, b, and c are integers, have such a structure. More precisely, the set of primitive SLsquare (Z) orbits of binary quadratic forms of given discriminant D has the structure of an abelian group. After this fundamental work of Gauss, there had been no progress for several centuries on discovering such structures in other classes of forms. Manjul Bhargava’s stunning work in his doctoral thesis, published as several papers in the annals of mathematics, shows how to address this question for cubic (and other higher degree) binary and ternary forms. The work of Bhargava, who is currently Professor of Mathematics at Princeton University, is deep, beautiful, and largely unexpected. It has many important ramifications and will likely form a theme of mathematical study at least for the coming decades. It is also sure to be a topic of discussion at the 2010 International Congress of Mathematicians in Hyderabad.

Why India Is A Hindu Rashtra

Originally from (http://psomax.blogspot.com/) with few modifications

Secularists just don't get it. They are simply not willing to admit that India is a Hindu country. The point is so obvious that one
doesn't know where to begin 'proving' it.Firstly, it's a matter of simple numbers. India is 80.5% Hindu. Muslims and Christians are 13.4% and 2.3% respectively. That alone should be enough to clinch the case. But there is more.

Throughout India's history, Hinduism has been the land's dominant tradition. In fact, for a long time it was the only tradition.

India's history began with the Indus Valley Civilisation (3000 BC –1700 BC). Elements from this civilisation were absorbed into the way of life that developed during the Vedic Period (1500 BC – 500 BC), forming what we today know as Hinduism.

The first alien faith (Christianity) arrived in India only in 50 AD, when we had been a purely Hindu civilisation for 3000 years! (And how much headway was made by this faith can be judged by the fact that after 2000 years of missionary activity and 200 years of European rule, it is still a mere 2% of the population) The second alien faith (Islam) arrived in India in 712 AD. So Islam has been in the subcontinent for only 1300 out of 5000 years of our history.

Secularists disagree with the above argument, saying that India was a "multi-religious" civilisation long before Christianity and Islam arrived. They point to Jainism and Buddhism. What they don't realise is that Jainism and Buddhism are not separate religions. These are just two of the many schools of thought (or 'sects') within Hinduism. Mahavira and Buddha did not teach anything fundamentally different from the Vedas. Their revolt was against the empty ritualism and ill practices that Vedic religion had degenerated into, not against the Vedas themselves. With their emphasis on concepts like satya and ahimsa - which are what the Vedas also teach - they succeeded in restoring Sanatana Dharma to its true spirit (A similar revival can be
seen in the Upanishads during the same period, albeit at the intellectual level). The teachings of Guru Nanak 2000 years later
should also be seen in the same light. Thus Jainism, Buddhism and Sikhism are just different streams (and such beautiful streams!) flowing into the great river of Sanatana Dharma. Yet secularists like Amartya Sen insist on saying that Ashoka, arguably India's greatest ruler, was "a Buddhist, not a Hindu".

A nation is defined by its culture. Secularists want us to believe that a piece of land + some people + some laws are enough to make a nation. This is a dry and lifeless idea of nationhood. It is incapable of inspiring love, loyalty, pride and a sense of belonging – all of which make life meaningful and give us a reason to sacrifice for the common good.A nation is above all a cultural entity. This is especially true of India. And Indian culture is basically Hindu culture. This is reflected in our greatest achievements in art, architecture and literature. The Vedas, the Ramayana, the Mahabharata, the great
temples of South India, the plays of Kalidasa (the list is endless) -which are the wonder of the world - are all essentially Hindu. It is also reflected in the lives of the greatest men produced by this land. From Buddha and Shankara down to Swami Vivekananda and Mahatma Gandhi,they are all known primarily for their contribution to Hindu thought and their upholding of Hindu values.True, other religions have also made their contributions. Islam, Hindustani classical music among other things. But these achievements were the result of foreign cultures mixing with the foundational culture, like rivers merging with the mighty ocean. And there can be no doubt what that foundational culture is. Secularists talk about composite culture.They refuse to recognise that we have a 'composite culture' if one want to call it is precisely because of Hinduism's tolerant and assimilative nature. Honesty lies in giving credit where it is due.

India is the oldest surviving civilisation in the world. The older civilisations (Sumer and Egypt) vanished long ago. Many younger
civilisations (Greece, Rome, Persia, etc) also arose and fell. But India is still standing after 5000 years – despite being subject to
countless invasions, conquests and long periods of foreign rule. How is this possible? How can a nation survive such severe trials? What gave it the strength to go on through its dark periods? What held it together through all the ups and downs of history? The answer is Sanatana Dharma.

Secularists say there was no India before 1947. They say the British made us a nation. I repeat: our nation is first and foremost a cultural entity. India may not have seen much political unity in its long history, but there was no lack of cultural unity. This land bound by the seas, the Himalayas and the Indus was one cultural unit. Its people may have followed different sects and worshipped different gods, but they all shared certain common core beliefs (such as karma and moksha). Irrespective of where they lived, they made pilgrimages to holy places scattered all over the country. The great epics were treasured in every home across the length and breadth of the land. It is not a coincidence that Shankara set up his four mutts in the four corners of India.

Thus it is Hinduism that binds this country. Secularists talk only about India's (and Hinduism's) diversity. They do not see the unity underlying this diversity. What unites us is much more than what (apparently) divides us. They say Hinduism is just one of the many religions in India. They are wrong; it is the first among equals. They say India is a salad bowl and Hinduism just one of the ingredients in the bowl. They are wrong; Hinduism is the bowl.

The case is overwhelming. Yet secularists refuse to accept the obvious truth. What could be behind this cussedness? A desire to be broad minded, open, universal and tolerant? But those are exactly the values Hinduism stands for! There is no contradiction between asserting India's Hinduness and wanting India to continue as a tolerant,pluralist society. Secularists say declaring India a Hindu country would undermine the secular nature of our Constitution. They don't see that a secular state is perfectly consistent with Hinduism. In Hindu India the state was always secular; there was never such a thing as a Hindu state.
Whichever sect a king belonged to, all his subjects had complete freedom of belief and worship. He did not discriminate among his subjects based on their sects; all subjects were equal; all sects were respected. Thus when the founding fathers of our Republic made India a secular state, they were not importing anything from the West (never mind what Nehru thought). They were merely staying true to Hindu tradition. Incidentally the Constitution did not originally contain the word 'secular'. The founding fathers saw no need for it. A secular state was given. It needed a cynical politician like Indira Gandhi to insert the word into the Preamble. How unnecessary!

Secularists say declaring India a Hindu country would pave the way for a Hindu theocracy. A Hindu theocracy is not possible because Hinduism is not an organised religion. Secularists are unwilling to admit that Hinduism – with its tolerance and catholicity – is responsible for India being a multi-religious country today. If India is a vibrant, pluralistic democracy, it is mainly because of its Hindu majority.Thus accepting India's Hinduness will only strengthen, not weaken, our secular polity.

"Why make such a fuss over this issue? Why not just maintain a polite silence? All this talk of Hindu rashtra may offend some people. Why is it so important that we openly proclaim this truth?" Because identity is important. It is important to know who we are and where we come from. Every individual is special. Who you are is your unique gift to society. What is true for individuals is also true for civilisations. Every civilisation has something to give to the world - which it can give only if it knows its true nature. A civilisation that loses its sense of identity is of no use to the world.

Yes, it is important to be welcoming and hospitable to others. But you can be welcoming and hospitable only if you have a house of your own. Identity is that house. Gandhiji said, "Keep your doors and windows open. Let the winds come in from all over the world. But do not be blown off your feet." Note that he said, "Keep your doors and windows open". He did not say, "Break your walls down". Just as we need four walls and a roof to protect us from heat, cold and rain, we need an identity to survive in the real world, in the world of culture and politics.

"Who am I?" is the most fundamental question a human being faces. Our civilisation gives us the answer to this question at two levels: the ultimate and the immediate. The answer at the ultimate level is "I am Brahman" |अहं ब्रह्मास्मि|. And the answer at the immediate level is "I am a Hindu". To give up lower identity for higher one is what our tradition taught us. But all identities are necessary at different point of time in life.

त्यजेदेकम कुलस्यार्थे | ग्रामास्यार्थे कुलम त्यजेत|

ग्रामम जनपद्स्यार्थे | आत्मार्थे प्रिथिवी त्यजेत|

India’s Scientific Heritage: Measurement of time
By Suresh Soni

http://www.organiser.org/dynamic/modules.php?name=Content&pa=showpage&pid=202&page=22

In India, sages contemplated on this and perceived it. Describing the condition before the creation of the universe, the Naasadiya Sookta of the Rigveda says that there was no truth or untruth, neither atom nor leisure. Then what was there? There was neither death nor immortality, neither day nor night. At that time, there was an element with the power of pulsation.

Darkness was enveloped with the darkness before creation and there was an element which had the power or the strength of penance. It was first the power of the effect of desire that the equilibrium was shattered and the universe was created from the unexpressed state. And the journey of time also began. This is how the journey of the time moves on along with the universe.

Defining this, Sages have described it as ‘kalayati sarvaani bhootani’ that is one that drives the entire universe or creation. It is also said that this universe is made once and then, gets destroyed, this is not the end. The cycle of birth and death, creation and destruction goes on. The giant wheel of time goes on with its creation, position, change and destruction. The poets of India and the west have described the all-eclipsing form of time alike. Kshemendra, a renowned poet of India has expressed his views thus-

Aho kaalasamudrasya na - lakshyante atisantataah
Majjantontaranantasya - yugaantaah parvataa iva.

“There is no such interval as compression in the ocean of time; huge mountains like massive ages come to submerge into it.” Octavia Paz, the poet who won the Nobel Prize in 1990, has, in his poem ‘Into the Matter’, described the all engulfing nature of time in the following words-

A clock strikes the time
Now it’s time
It is not time now, not it is now
Now it is time to get rid of time
Now it is not time
It is time and not now
Time eats the now
Now it is time
Windows close
Walls closed doors close
The words go home
Now we are more alone…….

The shortest as well as the largest unit of time has been described in our country.

There is a reference to this in the Shrimad Bhagwad Mahapurana. King Pareekshit asks Sage Shukdev what is time? What are its minutest and greatest forms? The reply that the sage gives is amazing because in today’s modern age, we know that time is an abstract element. We know it because of the incidents that occur. Thousands of years ago, Sage Shukdev had said, “The changing of subjects is the form of time. The element of time expresses itself through it (change). It expresses itself through the unexpressed.”

Measurement of Time
The minutest part of time is the atom and the greatest is Brahma Age. Explaining it in detail, Sage Shuk gives its various measures-

2 paramaanu - 1 Anu
3 Anu - 1 trasrenu
3 trasrenu - 1 truti
100 truti - 1 vedh
3 vedh - 1 lav
3 lav - 1 nimesh
3 nimesh - 1 kshan (moment)
5 kshan - 1 kaashthaa
15 kaashthaa - 1 laghu
15 laghu - 1 naarikaa
2 naadikaa - 1muhoort
30 muhoort - 1 day-night
7 day-night - 1 week
2 weeks - 1 fortnight
2 fortnight - 1 month
2 months - 1 ritu (season)
3 ritus - 1 ayan
2 ayans - 1 year

According to the calculations of Sage Shuk, there are 3,28,05,00,000 paramaanu of time in a day and night and 86,400 seconds in a day and night. This means that in its minutest measure, one paramaanu of time is equal to 37968th part of a second.

In A 231 of the Moksh Parva in the Mahabharata, time has been calculated as under-

15 Nimesh - 1 kaashtha
30 kaashthaa - 1 kala
30 kala - 1 muhoort
30 muhoort - 1day and night

There is a slight difference between the two calculations. According to Sage Shuk, there are 450 kaashthaas in a moment and according to the Mahabharata, there are 900 kaashthaas in a moment. This implies the different methods of calculations.

These are the unites for ordinary time calculations. But to measure the age of the universe or the changes therein, bigger units will be required. That measurement unit is yug.

Kaliyug - 432,000 years
2 Kaliyug Dwaparyug - 864,000 years
3 Kaliyug Tretayug - 1296,000 years
4 Kaliyug Satyug - 1728,000 years
The four ages together make a Chaturyuge - 4320,000 years
71 Chaturyugis make a Manvantar - 306720,00 years
14 Manvantaras along with 15 Satyugs as a part of the dusk make up a kalp that is - 4320,000,000 years

One kalpa means one day of Brahma. One night of His is equally long. One Brahma lives for 100 years and when one Brahma dies, it is Lord Vishnu’s nimesh (blinking of the eye), and after Vishnu, the age of Rudra starts. He is himself a form of kaal and is, therefore, eternal. That is why time is said to be endless.

After reading this description of Sage Shukdev, a thought that comes to mind is that this description is fantastic imagination and an intellectual game. What is the significance of such things in today’s scientific age? But this is not fantastic imagination. It is related to astronomy. India’s calculations of time were made on the basis of a minute study of the speed and changes in the astronomical bodies which means solid scientific truth; whereas in the calculations of the Anno Domini era prevalent today, the only scientific thing is the fact that the year is based on the calculations of the time that the earth takes to revolve around the sun. Otherwise, there is no relation between the calculations of the months and days and astronomical speeds.

STRUGGLE IS OVER HISTORY, NOT BRICK AND MORTAR

Dr. N.S. Rajaram (mathematician, linguist and historian)

The Ayodhya dispute

The Ayodhya dispute is over four hundred and fifty years old. It came to head on December 6, 1992 with the demolition of the structure known as the Babri Masjid (Babar's Mosque) by Hindu activists. This event has been seen as marking a watershed in modern Indian history. Some like the British writer V.S. Naipaul see it as an event marking the birth of a new historical awareness on the part of the Hindus; while others, calling themselves the 'Secular Forces' — actually little more than a motely mix of Leftist academics and politicians, and right-wing Muslim leaders and the clergy — see it as the beginning of the transformation of India into a Hindu theocratic state.

I see Ayodhya as the symbol of the emergence of the Indian Civilization — more specifically, the Hindu Civilization — from the grip of alien imperialistic forces and their surrogates that have tried to hold on to their privileged positions by suppressing the legitimate national and cultural aspirations of the Hindus. In this they have tried whitewashing the record of vandalism by Medieval Islamic rulers. This is what brought together this seemingly ‘modern’ and 'Westernized' Leftist intellectuals and right wing Islamists with their roots firmly in Medieval history and tradition. Koenraad Elst calls this whitewashing of history 'Negationism', more particularly 'Jihad Negationism'.

Negationism: Accepting the Islamic version of history

The present volume is only peripherally about Negationism. It is in the main a concise summary of the latest evidence on the Temple-Mosque controversy based on the primary sources including recent archaeological finds. I found it necessary to prepare this volume because there is still much confusion in the minds of many Indians about the existence of a Rama temple and its destruction by Babar in 1528. Many educated Indians still believe that there are some doubts about the historical question; many honestly believe that no temple was ever destroyed by Babar because he was tolerant towards the Hindus. (Towards the end, I have included a brief discussion of Babar’s famous work Baburnama to give an idea of what he was really like.)

This view, while a tribute to the effectiveness of negationist propaganda, is not a true representation of facts. In reality there can no doubt about either the existence or the destruction of a Rama temple by Babar at Rama Janmabhumi. What 'controversy' there is, is a modern concoction, the result of a massive disinformation campaign by 'Secularist' scholars, politicians and a large segment of the English language press. What is more important is that this happens to be part of a larger agenda of denying altogether the destruction of any Hindu temples by Muslim rulers — a step towards whitewashing the record of Islam in India. This is what Elst has called Negationism in his remarkable book Negationism In India: Concealing the Record of Islam. The reader will be the best judge of the facts upon reading the material presented in this volume.

A point that I wish to emphasize: any effort aimed at understanding the history leading up to the Ayodhya demolition must be careful not to view the events of December 6, 1992 in isolation, ignoring the thousand year history leading up to it. This would cause one to lose sight of the single most important historical theme in India today: the ongoing struggle between the two versions of history — the nationalistic and the imperialistic. Those calling themselves 'Secularists' in the Ayodhya dispute are representatives of defunct imperialisms — the Islamic and the European. What they fear most is the loss of their privileges following the rise of nationalism. This is the real battle over Ayodhya

The negationist version of Indian history means accepting the Islamic view of history — to wit, that the history of any place begins with its Muslim takeover; nothing that happened before is of any account. This is how Muslims view the history of all the conquered lands — from Egypt to Iran and even Pakistan. They have been defeated in their purpose to impose this version of history on India also. The struggle over Ayodhya is but a facet of this larger struggle.

This is best understood by recognizing that there are really two Ayodhya disputes. The first is over the right of possession to the site known since time immemorial as Rama Janmabhumi. The second is over the version of history to be imposed on the people of India today. The beneficiaries of defunct imperialisms — Islamic and the Eurocentric — are using the first dispute as a diversionary tactic to draw attention away from their real concern; their real concern is the unraveling of an imperialistic version of history highly advantageous to themselves. As surrogates of past imperialistic movements, they have also been its main beneficiaries. Once the true history comes to the fore, it will mean the end of their privileged existence.

To achieve their goal, the agents (or surrogates) of imperialisms have found it necessary to preserve and protect their (negationist) version of history. No imperialism can succeed unless the subject people are made to forget their history. This is what Islam did to Arabia, Egypt, Iran and Afghanistan; this is what Christianity did to Europe and the Americas; and this is what Mao also did to China, and later Tibet. And this is what the Secularists would like to see happen in India also. Destruction of history is the goal of every imperialism. Speaking of imperialisms and their specially crafted language, more specifically Islam, Sita Ram Goel observes:

... every language of imperialism divides human history into two sharply separated periods — an age of darkness which prevailed before the birth of an incomparable person, and an age of light which followed thereafter. The entire past history of every nation preceding the age of darkness is painted so black that nothing in which the nation can take pride is left unscathed. [This 'incomparable person' is Prophet Muhammad in the case of Islam, and Karl Marx in the case of the Secularists. So it is essentially a personality cult. Such cults were built around ‘incomparable persons’ Stalin and Mao also.]

The Secularists see India's indigenous Hindu Civilization as the dark force whose entire history should be blackened beyond redemption and ultimately effaced, to be replaced by its own Age of Secular Light. The first step is to coin a derogatory term for it — 'Hindu Communalism' (or Kaffir Communalism). They see India as an impure land plagued by pluralistic Hinduism that awaits Secularist cleansing. This is the Secularist version of the Islamic concept of Dar-ul-Harb and Dar-ul-Islam. Their version of the Ayodhya dispute which seeks to erase a thousand years of history leading up to December 6, 1992 is part of this enterprise.

End to Negationism

This again highlights the two Ayodhya disputes: the first ancient and historical, the second recent and ideological. To understand the second — the 'real dispute' so to speak — we must perforce study the first. To this end, I will present here the essential facts of history relating to the Ayodhya dispute. Once these are understood, readers will be in a better position to appreciate the real struggle that Ayodhya represents.

I recognize that much of what I have written here will make for unpalatable reading for many Muslims. But history is history, whether we like it or not. Also, no one is asking for revenge or retribution for past crimes. Nor has anyone the right to object to another's belief, as long as that belief remains personal. All one is asking is that Negationism must stop, so a true history can come to the fore. Above all, we cannot expect the Hindus to accept the Islamic view of history — that their civilization had engulfed India in a Dark Age to which light came only with the arrival of Islamic invaders. Nor can Muslims expect the Hindus to accept their version that nothing that happened at Ayodhya before the arrival of Babar is of any account, let alone their theology that there was no history before the arrival of Islam.

This is in the interests of all concerned — not just the Hindus. Communal harmony in India is an unattainable goal as long as one side keeps insisting on whitewashing its own record, while blaming the victims for all the problems. And the victims of such propaganda will never rest content until they feel their case has been justly treated. Here is where the Secularists have done immense harm to the cause of communal harmony in the name of 'secularism' — whitewashing Jihad Negationism, while heaping abuse and blame on the victims.

This is not a new or recent development. Within four years of Indian independence and the Partition, the late K.M. Munshi had warned Nehru of the dangers of this less than honest stand on secularism. In a now famous and remarkably prophetic letter, Munshi, one of the foremost constitutional lawyers of the day wrote Nehru:

In its [i.e. secularism's] name, again, politicians adopt a strange attitude which, while it condones the susceptibilities, religious and social of the minorities, it is too ready to brand similar susceptibilities in the majority community as communalistic and reactionary. How secularism sometimes becomes allergic to Hinduism will be apparent from certain episodes relating to the reconstruction of Somnath Temple.

These unfortunate postures have been creating a sense of frustration in the majority community. If, however, the misuse of this word 'secularism' continues, ... if, every time there is an inter-communal conflict, the majority is blamed regardless of the merits of the question, ..., the springs of traditional tolerance will dry up.

While the majority exercises patience and tolerance, the minorities should adjust themselves to the majority. Otherwise the future is uncertain and an explosion cannot be avoided. (Emphasis added.)

Prophetic indeed, written forty years before the explosion at Ayodhya! And this has gone on for nearly fifty years. It looks as though nothing has been learnt by the Secularists and their allies. The politicians too keep on promising the impossible to the Muslims in the hope of garnering their votes. If this goes on much longer, more explosions like Ayodhya become all but inevitable.

The result of this has been most unfortunate; it has turned the traditionally tolerant Hindus into a majority community with a persecuted minority complex — making them believe that they are second-class citizens in their own country. (This has now been aggravated by the aggressive behavior of Christian missionaries caught in a millennium frenzy.) This is a dangerous development that bodes ill for the minority community, and for the country at large. In this the culprit is Jihad Negationism, and the suspicion and hatred that it breeds.

What are we to do about it? In this regard, one can learn a valuable lesson by looking at European history. The record of Christianity in Europe and the Americas is no less blood-soaked than the record of Islam in India. But there are no 'Crusade Negationists' or 'Inquisition Negationists' in Europe comparable to the Jihad Negationists in India. This has allowed communal harmony to prevail in Europe.

The lessons for India are clear, she must come to terms with her history. A similar situation prevailed in the United States over the question of slavery. There was no shortage of negationists who argued that slavery was a natural law that contained much good. But Abraham Lincoln, one of the greatest men of modern times, would have none of it. He told Americans to face up to their history:

Fellow citizens, we cannot escape history. ... No personal significance or insignificance can spare one or another of us. The fiery trial through which we pass will light us down in honor or dishonor to the last generation.

Truer words were never spoken. Its message is clear: for peace and harmony to prevail in India, Negationism must end. Indian history must be freed from the shackles of its imperial surrogates acting in the name of 'secularism'.

It is time now to look at the Ayodhya dispute against the background of this brand of 'secularism' and the conduct of its votaries.

Lying secularly

Recently I came across a website by name 'The campaign to stop funding hate' http://www.stopfundinghate.org/. The people behind this work are mostly US based intellectuals who run many NGOs back home and collects money from US for developmental projects in India. The effort is genuine and we need many more such NGOs so that NRIs and foreign funding agencies can be effectively utilized for nation building. But along with this noble work these groups are also involved in proving that they are the only authentic Indian and rest all are sectarian groups not entitled to collect money for Indian people. In doing so they even forget that they are tarnishing image of Indian majority culture which they claim to help with their funding. As the name goes by, the people involved in this project literally 'HATE' anything related to hinduism and want to believe us that India is not a birthplace of hinduism and the modern India has a composite culture evolved by mixing indigenous and invaders alike.

As a hindu, I even agree to diversity in ideology. Ultimately we all are looking forward to build a nation with our own ideas. There is nothing wrong in it. Let people in and outside of India decide our sincerity and they will support us in every possible way. But by taking our local matters to international stage we as a nation is at loosing end. When these people blame RSS for spreading hate in India against minorities, they are doing same outside India against majority. Money alone cannot do all. Devoted people for implementations of the ideas are essential. RSS is running more than 70,000 projects in India and not at all dependent on foreign funding. http://video.google.com/videoplay?docid=-1013666219441914390

Let us accept the fact that RSS is and will not be affected by this hate mission but certainly the proponents in the eyes of society. RSS believes that many more efforts are required to build the nation and everyone can participate in his/her capacity with his/her own ideas.

Following are few thoughts may be helpful to understand RSS better if one wish to:

- According to Darwin, only fit survives through evolution. Our traditions dates back prehistoric period according to UNESCO (Rigveda: first written document). Is it not worthwhile to understand what is so special about us when Egypt, Roman, Babylonian, Greek, Persian,Syrian and many more cultures are non existent today?

- We evolve through time, our thoughts changes as per the situation, experiences. It is also applicable to society, culture and even to organization. Most flexible culture sustains. Rigid one remains only part of history. Flexible culture has more probability of incorporation of situational customs. Is it justifiable to cut the head for headache instead of treating for pain?

- If hindu majority is violent all along the history then is it possible to see the mosaic structure of our present society?

- Hindutva of RSS is politically not convenient for its ideological opponents and hence they brand this is as not original hinduism. Our all deities are with weapons and teaches us to use them whenever necessary. For hindus, who believe in coexistence, it is always the last option and of course not desirable at all. But today we forgot that such option exist and any reaction of hindu soceity is seen as fascism and branded as 'majority communalism'. Giving reaction for any religious and secular problem is reserved for others and not for hindus. An eye for eye will certainly make whole world blind but at the same time this is also true that otherwise the world will have only one type of eyes which is evident by 54 muslim and over 120 christian countries where other 'eyes' were wiped out on the surface of this planet in the name of 'jihad' and 'crusade'.

- Organized religions always prospered through political system hence in west they have to use the word 'secular' and devise a system through which chruch was kept out of government. Hinduism never flourished through king like christianity as the core values of hinduism does not belong to materialistic world. The concept of secularism is not applicable in Indian scenario at all.

- Hindu rashtra for RSS is never a theocratic state which is also confirmed by 1995 landmark judgement of supreme court that says hinduism is a way of life and not only set religious rituals. RSS is well aware that 99% of non-hindus of todays India are nothing but converts, (which itself totally non spiritual concept on its own). Just by changing the mode of worship one cannot forget his forefathers or their traditions. RSS expects from non-hindus to be part of this hindu way of life along with following their new mode of worship. Whenever there is conflict between their new mode of worship and traditions of land (e.g. reciting vande mataram, sarsawti vandana etc) they have to sort out within their community and mould it according to later. That is the only way to be part of main stream of nation according to RSS which is not at all extravagant as everywhere in the world it was the same thing

Ex: 1) Kaba was the temple with thousands of idols which is later converted to holy mosque, 2) Christmas in europe is coupled with pagan practises ('weihnachten' in germany 'jul' in sweden etc, 3) Indonesia has 'ganapati' on their currency and 'ramayan' as national book, 4) Buddhism in japan and china is modified version which is assimilated with Confucianism, Shintoism, and Taoism. One can find many more such cases all over the world.

- If 18% of muslims and 7-8% other minorities are always kept out of main stream of society for pity political reasons we as a nation will be always facing internal conflicts. Where ever hindus are in minority we can see that part of country is cut away physically like pakistan and mentally like north east. If we want to move as cohesive unit in future we need to search for unity among the superficial diversity. We have that unity in the form of 'hindu way of life'. We have to promote it which was there in past. Unfortunately we today are more interested in showing how we different and/or how backward than each other to get reservations. Hinduism too has its limitations and also need to be addressed in changing time which is not possible by alienating us from it.

One may be having genuine differences of opinion about RSS views. Still we can all work with our own ideas for India without pointing to each other. Let's put historical facts as it is in front of the new generation without our own flavour in it and let them decide which path to choose.

But ultimately until the hunters are history writers we will have glorification of hunters and not of lions.We have to see the history from our perspective and not the invaders.