Mathematicians from India

"Like other social media platforms, Wikipedia has evolved into an echo chamber where the user is presented with only one type of content instead of being shown a balanced narrative. This disinformation is powerful since the articles are written in an academic style and users do not see other sources that disagree with the article.... Some editors of Wikipedia are failed academics with demonic energy who wish to conquer anonymously what they were unable to do in their normal careers. And spending much of their working life editing Wikipedia articles and by the use of multiple anonymous handles they have obtained administrative status which entitles them to block opposing views. The anonymous persona of the editors and the low stakes have made Wikipedia politics much more vicious than real politics."

"Once scientists and scholars invest parts of their career in support of a paradigm, it becomes a sort of a self-betrayal to abandon it."

"Men and women in their mutual attraction are driven to the very emptiness they are trying to avoid."

"The essence of the Vedas is a narrative on who the experiencing self is. Ordinary science informs us of the relationships between objects and also their transformations. But the Vedas say that this ordinary science leaves out the self who observes these objects."

"Beauty takes us to a space that is ineffable, a place of secrets."

"Indian Anglosphere indifference to #KashmiriPandit genocide & expulsion is a shameful episode of recent history. Look at the Yazidis; their genocide and sex-slavery is not rationalized. The Indian Anglosphere lives in a cocoon of self-hate, self-deception and brazen mendacity."

"The best paradise is the paradise we are exiled from."

"What is the chance that one can roll up the sky like a hide?"

"The world is a game of information and paradox."

"I have so much of desire that desire itself is my fulfillment."

"People embrace false magical theories in the hope something good will come out of them. In the most extreme of these, good comes out of them only at the end of this life, in paradise."

"Man is a mimic animal, happiest acting a part, needing a mask to tell the truth."

"Gods have many faces."

"The body is like the wife to the spirit. The two must cohabit to create new forms, but their pleasures rarely coincide."

"When the mind grasps the universe, the senses retreat."

"History is scraps of evidence joined by the glue of imagination."

"Ritual is intimately connected with the mask, either in the wearing that hides the true face, or in the adoption of a public face."

"Modern life alienates us from Nature, even our own."

"If the heart sorrows over physical loss, the spirit rejoices over hope of understanding."

"Europe has resurrected its pagan gods."

"The dance of the peacock attracts not only the peahen but also the human."

"The clash of civilizations is nothing but a clash of different myths."

"One is not a single self, although there is some common thread holding together disparate incarnations."

"If chess is about decisive victory by vanquishing the enemy by taking the fight to the place where the king is located, weiqi is about consolidation of territory."

"If social media can bring the sense of freedom, it can also bind people into delusional cults."

"The idea of consciousness requires not only an awareness of things, but also the awareness that one is aware."

"Since language is linear, whereas the unfolding of the universe takes place in a multitude of dimensions, language is limited in its ability to describe reality."

"There is nothing as uplifting and inspiring as the Upanishads."

"A culture is like a lens through which people construct their world."

"I got to visit the ruins of the Mārtaṇḍa Temple on the Mattan Karewa, built by Emperor Lalitāditya, many times. Built of stone, it is characterized by the simplicity of its conception: it is rectangular in plan, consisting of a maṇḍapa and a shrine. Two other shrines flank the maṇḍapa. It is enclosed by a vast courtyard by a peristyle wall with 84 secondary shrines in it. The columns of the peristyle are fluted. Each of the 84 niches originally contained an image of a form of Sūrya. The number 84, as 21×4, appears to have been derived from the numerical association of 21 with the sun. It must have looked like a jeweled treasure on the plateau over Anantnag."

"An evil wind destroyed that old Kashmir. I had a premonition of this disaster on a visit thirty years ago in the averted eyes, in the barely concealed rage in people I knew. It was a rage unlike the one that sprouts from a personal wrong. It was like a fog that hung over the place in heavy layers, covering everyone, even those who would normally be happy in their own world."

"This campaign of terror was nowhere as widespread and sustained as in Kashmir. The Hindus fled their homes and took refuge wherever they could, with their lives shattered forever. As refugees, hundreds of miles away from their homes, they were housed in one-room hovels in Jammu. I was invited to give a speech at a meeting called by young Kashmiris in Jammu in December 1991 to reflect on what should be done. I got an opportunity to see first-hand the condition of the refugees, and it was heartbreaking."

"The youth established an organization called Panun Kashmir and they declared as their goal the unrealistic idea of a homeland for the Hindus within the valley. This was more a cry of helplessness than anything else."

"If n is any positive quantity shew that \frac 1{n} > \frac 1{n+1} + \frac 1{{(n+2)}^2} + \frac 3{{(n+3)}^3} + \frac {4^2}{{(n+4)}^4} + \frac {5^3}{{(n+5)}^5} + \dots Find the difference approximately when n is great. Hence shew that \frac 1{1001} + \frac 1{1002^2} + \frac 3{1003^3} + \frac {4^2}{1004^4} + \frac {5^3}{1005^5} + \dots < \frac 1{1000} by 10^{-440} nearly."

"Sir, an equation has no meaning for me unless it expresses a thought of GOD."

"I beg to introduce myself to you as a clerk in the Accounts Department of the Port Trust Office at Madras... I have no University education but I have undergone the ordinary school course. After leaving school I have been employing the spare time at my disposal to work at Mathematics. I have not trodden through the conventional regular course which is followed in a University course, but I am striking out a new path for myself. I have made a special investigation of divergent series in general and the results I get are termed by the local mathematicians as "startling". ...Very recently I came across a tract published by you styled Orders of Infinity in page 36 of which I find a statement that no definite expression has been as yet found for the number of prime numbers less than any given number. I have found an expression which very nearly approximates to the real result, the error being negligible. I would request that you go through the enclosed papers. Being poor, if you are convinced that there is anything of value I would like to have my theorems published. I have not given the actual investigations nor the expressons that I get but I have indicated the lines on which I proceed. Being inexperienced I would very highly value any advice you give me. Requesting to be excused for the trouble I give you. I remain, Dear Sir, Yours truly..."

"Ramanujan lived in a tiny hut in India. No formal education, no access to other works. But he came across an old math book and from this basic text he was able to extrapolate theories that had baffled mathematicians for years. … Ramanujan's genius was unparalleled."

"Paul Erdős has passed on to us Hardy's personal ratings of mathematicians. Suppose that we rate mathematicians on the basis of pure talent on a scale from 0 to 100, Hardy gave himself a score of 25, Littlewood 30, Hilbert 80 and Ramanujan 100."

"He began to focus on mathematics at an early age, and, at the age of about fifteen, borrowed a copy of G. S. Carr's Synopsis of Pure and Applied Mathematics, which served as his primary source for learning mathematics. Carr was a tutor and compiled this compendium of approximately 4000-5000 results (with very few proofs) to facilitate his tutoring."

"At about the time Ramanujan entered college, he began to record his mathematical discoveries in notebooks... Ramanujan devoted all of his efforts to mathematics and continued to record his discoveries without proofs in notebooks for the next six years."

"After Ramanujan died, Hardy strongly urged that Ramanujan's notebooks be edited and published. By "editing," Hardy meant that each claim made by Ramanujan in his notebooks should be examined. If a theorem is known, sources providing proofs should be provided; if an entry is known, then an attempt should be made to prove it."

"Ramanujan's approach to the theory of theta functions does not appear to have been influenced by any other writer."

"He was sent at seven to the High School at , and remained there nine years. ...His biographers say ...that soon after he had begun the study of , he discovered for himself "Euler's theorems for the sine and cosine (by which I understand the relations between the circular and exponential functions), and was very disappointed when he found later, apparently from the second volume of Loney's Trigonometry that they were known already. Until he was sixteen he had never seen a mathematical book of higher class. Whittaker's Modern Analysis had not yet spread so far, and Bromwich's Infinite Series did not exist. ...[E]ither of these books would have made a tremendous difference ..."

"Ramanujan did not seem to have any definite occupation, except mathematics, until 1912. In 1909 he married, and it became necessary for him to have some regular employment, but he had great difficulty in finding any because of his unfortunate college career. About 1910 he began to find more influential Indian friends, Ramaswami Aiyar and his two biographers, but all their efforts to find a tolerable position for him failed, and in 1912 he became a clerk in the office of the Port Trust of Madras, at a salary of about £30 per year. He was nearly twenty-five. The years between eighteen and twenty-five are the critical years in a mathematician's career, and the damage had been done. Ramanujan's genius never had again its chance of full development."

"It has not the simplicity and the inevitableness of the very greatest work; it would be greater if it were less strange. One gift it shows... profound and invincible originality. He would probably been a greater mathematician if he could have been caught and tamed a little in his youth; he would have discovered more that was new, and... of greater importance. On the other hand he would have been less of a Ramanujan, and more of a European professor, and the loss might have been greater than the gain... the last sentence is... ridiculous sentimentalism. There was no gain at all when the College at Kumbakonam rejected the one great man they had ever possessed, and the loss was irreparable..."

"The formulae... defeated me completely; I had never seen anything in the least like them before. A single look at them is enough to show that they could only have been written by a mathematician of the highest class. They must be true because, if they were not true, no one would have the imagination to invent them."

"I hardly asked him a single question of this kind; I never even asked him whether (as I think he must have done) he had seen Cayley's or Greenhill's Elliptic Functions. ... he was a mathematician anxious to get on with the job. And after all I too was a mathematician, and a mathematician meeting Ramanujan had more interesting things to think about than historical research. It seemed ridiculous to worry him about how he had found this or that known theorem, when he was showing me half a dozen new ones almost every day."

"He could remember the idiosyncrasies of numbers in an almost uncanny way. It was Littlewood who said that every positive integer was one of Ramanujan's personal friends. I remember once going to see him when he was ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavorable omen. "No," he replied, "it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways.""

"The years between 18 and 25 are the critical years in a mathematician's career, and the damage had been done. Ramanujan's genius never had again its chance of full development. ... a mathematician is often comparatively old at 30, and his death may be less of a catastrophe than it seems. Abel died at 26 and, although he would no doubt have added a great deal more to mathematics, he could hardly have become a greater man. The tragedy of Ramanujan was not that he died young, but that, during his five unfortunate years, his genius was misdirected, side-tracked, and to a certain extent distorted."

"In his insight into algebraical formulae, transformation of infinite series, and so forth, that was most amazing. On this side most certainly I have never met his equal, and I can compare him only with Euler or Jacobi."

"The formulae (1.10) - (1.13) are on a different level and obviously both difficult and deep... (1.10) - (1.12) defeated me completely; I had never seen anything in the least like them before. A single look at them is enough to show that they could only be written by a mathematician of the highest class. They must be true because, if they were not true, no one would have the imagination to invent them."

"His death is the saddest event in my professional career. It is not for me to assess Ramanujan's mathematical genius. But at the human level, he was one of the noblest men I have met in my life-shy, reserved and endowed with an infinite capacity to bear the agonies of the mind and spirit with fortitude."

"Srinivasa Ramanujan was the strangest man in all of mathematics, probably in the entire history of science. He has been compared to a bursting supernova, illuminating the darkest, most profound corners of mathematics, before being tragically struck down by tuberculosis at the age of 33, like Riemann before him."

"The number 24 appearing in Ramanujan's function is also the origin of the miraculous cancellations occurring in string theory. ...each of the 24 modes in the Ramanujan function corresponds to a physical vibration of a string. Whenever the string executes its complex motions in space-time by splitting and recombining, a large number of highly sophisticated mathematical identities must be satisfied. These are precisely the mathematical identities discovered by Ramanujan. ...The string vibrates in ten dimensions because it requires... generalized Ramanujan functions in order to remain self-consistent."

"A more recent example of anomalous creativity is the work of the great mathematician Srinivasa Ramanujan, who died in 1920 at the age of 32. His notebook, which was lost and forgotten for about 50 years and published only in 1988, contains several thousand formulas -- without proof, in different areas of mathematics -- that were well ahead of their time, and the methods by which he found the formulas remain elusive. Ramanujan himself claimed that the formulas were revealed to him in his sleep. Is it possible that the creative process draws upon the unconscious in a manner that cannot be explained in rational terms?"

"Ramanujan learned from an older boy how to solve cubic equations. He came to understand trigonometric functions not as the ratios of the sides in a right triangle, as usually taught in school, but as far more sophisticated concepts involving infinite series. He'd rattle off the numerical values of π and e, "transcendental" numbers appearing frequently in higher mathematics, to any number of decimal places. He'd take exams and finish in half the allotted time. Classmates two years ahead would hand him problems they thought difficult, only to watch him solve them at a glance. … By the time he was fourteen and in the fourth form, some of his classmates had begun to write Ramanujan off as someone off in the clouds with whom they could scarcely hope to communicate. "We, including teachers, rarely understood him," remembered one of his contemporaries half a century later. Some of his teachers may already have felt uncomfortable in the face of his powers. But most of the school apparently stood in something like respectful awe of him, whether they knew what he was talking about or not. He became something of a minor celebrity. All through his school years, he walked off with merit certificates and volumes of English poetry as scholastic prizes. Finally, at a ceremony in 1904, when Ramanujan was being awarded the K. Ranganatha Rao prize for mathematics, headmaster Krishnaswami Iyer introduced him to the audience as a student who, were it possible, deserved higher than the maximum possible marks. An A-plus, or 100 percent, wouldn't do to rate him. Ramanujan, he was saying, was off-scale."

"Ramanujan was an artist. And numbers — and the mathematical language expressing their relationships — were his medium. Ramanujan's notebooks formed a distinctly idiosyncratic record. In them even widely standardized terms sometimes acquired new meaning. Thus, an "example" — normally, as in everyday usage, an illustration of a general principle — was for Ramanujan often a wholly new theorem. A "corollary" — a theorem flowing naturally from another theorem and so requiring no separate proof — was for him sometimes a generalization, which did require its own proof. As for his mathematical notation, it sometimes bore scant resemblance to anyone else's."

"Ramanujan was a man for whom, as Littlewood put it, "the clear-cut idea of what is meant by proof ... he perhaps did not possess at all"; once he had become satisfied of a theorem's truth, he had scant interest in proving it to others. The word proof, here, applies in its mathematical sense. And yet, construed more loosely, Ramanujan truly had nothing to prove. He was his own man. He made himself. "I did not invent him," Hardy once said of Ramanujan. "Like other great men he invented himself." He was svayambhu."

"Graduating from high school in 1904, he entered the University of Madras on a scholarship. However, his excessive neglect of all subjects except mathematics caused him to lose the scholarship after a year, and Ramanujan dropped out of college. He returned to the university after some traveling through the countryside, but never graduated. ...His marriage in 1909 compelled him to earn a living. Three years later, he secured a low-paying clerk's job with the Madras Port Trust."

"Every positive integer is one of Ramanujan's personal friends."

"I read in the proof-sheets of Hardy on Ramanujan: 'As someone said, each of the positive integers was one of his personal friends.' My reaction was, 'I wonder who said that; I wish I had.' In the next proof- sheets I read (what now stands), 'It was Littlewood who said... '"

"Ramanujan's great gift is a 'formal' one; he dealt in 'formulae'. To be quite clear what is meant, I give two examples (the second is at random, the first is one of supreme beauty):p(4)+p(9) x+p(14) x^2+\ldots=5 \frac{\left\{\left(1-x^5\right)\left(1-x^{10}\right)\left(1-x^{15}\right) \ldots\right\}^5}{\left\{(1-x)\left(1-x^2\right)\left(1-x^3\right) \ldots\right\}^6} where p(n) is the number of partitions of n; ... But the great day of formulae seems to be over. No one, if we are again to take the highest standpoint, seems able to discover a radically new type, though Ramanujan comes near it in his work on partition series; it is futile to multiply examples in the spheres of Cauchy's theorem and elliptic function theory, and some general theory dominates, if in a less degree, every other field. A hundred years or so ago his powers would have had ample scope... The beauty and singularity of his results is entirely uncanny... the reader at any rate experiences perpetual shocks of delighted surprise. And if he will sit down to an unproved result taken at random, he will find, if he can prove it at all, that there is at lowest some 'point', some odd or unexpected twist. ... His intuition worked in analogies, sometimes remote, and to an astonishing extent by empirical induction from particular numerical cases... his most important weapon seems to have been a highly elaborate technique of transformation by means of divergent series and integrals. (Though methods of this kind are of course known, it seems certain that his discovery was quite independent.) He had no strict logical justification for his operations. He was not interested in rigour, which for that matter is not of first-rate importance in analysis beyond the undergraduate stage, and can be supplied, given a real idea, by any competent professional."

"He was eager to work out a theory of reality which would be based on the fundamental concept of "zero", "infinity" and the set of finite numbers … He sometimes spoke of "zero" as the symbol of the absolute (Nirguna Brahman) of the extreme monistic school of Hindu philosophy, that is, the reality to which no qualities can be attributed, which cannot be defined or described by words and which is completely beyond the reach of the human mind. According to Ramanuja the appropriate symbol was the number "zero" which is the absolute negation of all attributes."

"Srinivasa Ramanujan, discovered by the Cambridge mathematician G. H. Hardy, whose great mathematical findings were beginning to be appreciated from 1915 to 1919. His achievements were to be fully understood much later, well after his untimely death in 1920. For example, his work on the highly composite numbers (numbers with a large number of factors) started a whole new line of investigations in the theory of such numbers."

"Ramanujam used to show his notes to me, but I was rarely able to make head or tail of at least some of the things he had written. One day he was explaining a relation to me; then he suddenly turned round and said, "Sir, an equation has no meaning for me unless it expresses a thought of GOD." I was simply stunned. Since then I had meditated over this remark times without number. To me, that single remark was the essence of Truth about God, Man and the Universe. In that statement, I saw the real Ramanujam, the philosopher mystic-mathematician."

"The manuscript of Ramanujan contained theorems and propositions that Hardy classified in three categories: 1) important results already known or demonstrable, through theorems which Ramanujan was certainly not acquainted with; 2) false results (few in number) or results concerning marginal curiosities; 3) important theorems not demonstrated, but formulated in such a manner that presupposed views... which only a genius could have."

"Hardy... in vain, tried to convince him to learn classical foundations of mathematics and, in particular, the rigorous expositive method of mathematical demonstrations. Every time Hardy introduced a problem, Ramanujan considered it ex novo [new] applying unconventional reasoning which was sometimes incomprehensible to his fellow colleagues."

"The great advances in mathematics have not been made by logic but by creative imagination. The title of mathematician can scarcely be denied to Ramanajan who hardly gave any proofs of the many theorems which he enumerated."

"That Ramanujan conceived these problems, sometimes before anyone else had done so, with no contact with the European mathematical community, and that he correctly obtained the dominant terms in asymptotic formulas are astounding achievements that should not be denigrated because of his unrigorous, but clever, arguments."

"Ramanujan proved many theorems for products of hypergeometric functions and stimulated much research by W. N. Bailey and others on this topic."

"Almost certainly his best piece of work and one of the very best achievements in Indian Mathematics since Ramanujan."

"In India, there's lack of appreciation of the need to cross-examine data, the responsibility of a statistician."

"The spirit and outlook of 'Sankhya' will be universal, but its form and content must necessarily be, to some extent, regional. We shall keep the special needs of India in view without, however, restricting the scope of the journal in any way. We shall naturally devote closer attention to the collection and analysis of data relating to India, but we shall try to study all Indian questions in relation to world problems.... The study of modern statistical methods in its infancy in our country, and we do not expect to be able to achieve immediate results. We shall be satisfied if we can help by our humble efforts to lay the foundations for future work."

"He sometimes spoke of "zero" as the symbol of the absolute (Nirguna Brahman) of the extreme monistic school of Hindu philosophy, that is, the reality to which no qualities can be attributed, which cannot be defined or described by words and which is completely beyond the reach of the human mind. According to Ramanujan the appropriate symbol was the number "zero" which is the absolute negation of all attributes."

"Because demography is concerned with human affairs and human populatlons it is possible, in principle, to consider demography as a sub-field of many other subjects. It provided the scope of any particular subject-field like anthropology, genetics, ecology, economics, sociology, etc., and is defined in a sufficiently comprehensive manner. While not denying the possibility of considering demography as a sub-field of one or another subject, at least for certain special purposes, it is suggested that demography should be logically viewed as the totality of convergent and inter-related factors and topics which (although these could be, spearately, the concern of many difl'erent subjects like genetics and anthropology, sociology, education, psychology. economics, social and political affairs etc.) jointly, together with their mutual inter-actions, form the determinants as well as the consequences of growth (or decline), changes in composition, territorial movements, and social mobility of population in different geographical regions or in the world as a whole, at any given period of time, or over difl'erent periods of time. Such a view would supply an aggregative, inter-related, and mutually interacting system of all those factors which have any influence over, or are influenced by, demographic or population changes over space and time."

"Population in India is widely differentiated in ethnic composition, geographical and climatic conditions, social and cultural stratification, as well as by differences in economic status. Differential fertility therefore assumes a far more complex picture in India than anywhere in the world. Ethnic. geographical. socio-cultural and economic dilferences give a four-fold patterning with many complicated interactions. It is essential therefore to study different population groups separately."

"some evidence is available to indicate that, in India, an increase in the income of the poorer people leads to an increase in the size of the family; and also that this tapers off after a certain critical level of income is attained, and is followed by a reduction in the size of the family at higher levels of living When a sufficient number of people reach the critical income, there would be a gradual decrease in the average birth rate with further increase in income."

"The transformation of the advanced countries to their present stage has been brought about by the acceptance of a scientific and rational view of life and nature. The scientific view has already permeated in a large measure the administrative organizations of the advanced countries. The scientific revolution, the social revolution and the industrial revolution are three aspects of the modernization of every society; these three aspects may be distinguished but cannot be separated. The rate of economic growth in every country is determined both directly and indirectly by the rate of progress of science and technology; directly through the utilization of the results of research and development, and indirectly through institutional changes brought about by the increasing influence of the scientific out-look and tradition."

"India has a medieval and authoritarian structure of society and the tradition of science is not yet strong. The power of government officials is increasing as an inescapable result of the pervasive anthoritarian character of lndian society."

"In the absence of social awareness and appreciation of the scientific objectivity among sufficiently large number of civil servants or political leaders,the need of validity has not yet been accepted in the official statistical system in India. Ofcial statistics in India is treated as an integral part of the dministrative system which is regulated by the principle of authority. Approval of statistical estimates at a high level of authority is accepted as a bstitnte for validity in many ases there is continuing opposition to independent cross-hccks for the validity of the data. Officials have the feeling that two independent estimates, which might differ would be confusing and, in fact unthinkable; therefore independent cross-checks in statistics should be eliminated."

"Without the progress of equality and improvement in the level of living at least beyond the poverty line, for one quarter of the population of the world who live in South Asia, there would be grave repercussions on the rest of the world. The problem of the underdeveloped country is, in one sense, of greater concern to the advanced countries because international rivalries and tensions arise from the desire to establish spheres of influence over underdeveloped areas. The very existence of underdeveloped regions would he therefore a continuing threat to world security, and world peace. A quick transformation of the underdeveloped countries into industrialized economies would reduce the sphere of conflicting interests; and hence decrease the tension between East and West."

"We believe that the idea underlying this integral concept of statistics finds adequate expression in the ancient Indian work Sankhya in |Sanskrit the usual meaning is ‘number‘, but the original root meaning was ‘determinate knowledge’ in the Atharva Veda a derivative from Sankhyata occurs both in the sense of ‘well-known‘ as well as ‘numbered’. The lexicons give both meanings. Amarakosa gives Sankhya – vicarana (deliberation, analysis) as well as ‘number’; also Sankhyavan – panditah (wise, learned)."

"It would be, however, a fatal mistake to establish an expensive system of education on the model of the advanced countries which would have little relevance to local needs and would be beyond the means of the national economy. It is necessary to evolve a system, through experimentation and trial and success, which would be within the means of the national economy."

"[He} was one of Tagore's rare friends who did not place him simply on a high pedestal full of only aura and fame, but treated him as a lively intellectual and affectionate companion."

"Just as Tagore sought to bring humanity closer through Visva-Bharati or his one-nest-world university at Santiniketan, Prasanta Chandra strove to use the ideal of humanism through statistics."

"The 'Mahalanobis Era' in statistics which started in the early twenties has ended. Indeed it will be remembered for all time to come as the golden period of statistics in India, marked by intensive development of a new technology and its applications for the welfare of mankind."

"C.R.Rao quoted in }Prasanta Chandra Mahalanobis""

"...l have been deeply struck by his broad and comprehensive approach to National Development and his astonishing energy. He is full of ideas and it is always a pleasure to discuss any subject with him."

"I need hardly say that I refer to the emergence of a statistically competent technique of Sample Survey, with which I believe Professor Mahalanobis name will always be associated."

"What at first strongly attracted my admiration was that the Professor‘s work was not imitative….The experience of India will serve as a guidance and as an example worthy of imitating."

"Seng in "Professor P.C. Mahalanobis and the Development of Population Statistics in India""

"No technique of random sample has, so far as I can find, been developed in the United States or elsewhere, which can compare in accuracy or in economy with that described by Professor Mahalanobis."

"Everybody knows him as the founder of the Indian Statistical Institute, the architect of the Second Five Year Plan, a close associate of Rabindranath Tagore and as one who had richly contributed to the social, cultural and intellectual life in Bengal. All those in the statistical profession were aware of his deep contributions to statistical theory, his efforts in providing a sound database to the Indian economy, and the part he played in placing India not far from the centre of the statistical map of the world. Those who have been closely associated with him have witnessed the indomitable courage and tenacity in fighting opposition for a good cause and clearing obstacles for propagating right principles."

"l have always noticed how you are always capable to maintain objectivity in your judgement about people and l have always recognised that to be a great quality in you."

"What you have written after analyzing everything connected with my achievements and fame is altogether correct."

"l have liked your article very much. The way you have narrated the history of my humanism in an evolutionary perspective has made this aspect of mine clearer even to me."

"If Mahalanobis had done nothing else, if he had only founded Sankhya, the Indian Journal of Statistics, even so his contribution to science would have been outstanding and memorable. Sankhya is an international journal in the sense that it receives contributions from statisticians and probabilists the world over; international as in the sense of maintaining a standard comparable to the best in the world. And this has been from the very beginning. This is something that cannot be said of many scientific journals in the country"

"The globe of the Earth stands supportless in space... Just as the [spherical] bulb of a Kadamba flower is covered all around by blossoms, just so is the globe of the Earth surrounded by all creatures, terrestrial as well as aquatic."

"When sixty times sixty years and three quarter yugas (of the current yuga) had elapsed, twenty three years had then passed since my birth."

"Just as a man in a boat moving forward sees the stationary objects (on either side of the river) as moving backward, just so are the stationary stars seen by people at Lanka as moving exactly towards the west. (It so appears as if ) the entire structure of the asterisms together with the planets were moving exactly towards the west of Lanka, being constantly driven by the provector wind, to cause their rising and setting."

"100 plus 4, multiplied by 8, and added to 62,000: this is the nearly approximate measure of the circumference of a circle whose diameter is 20,000."

"In Indian astronomy, the prime meridian is the great circle of the Earth passing through the north and south poles, Ujjayinī and Laṅkā, where Laṅkā was assumed to be on the Earth's equator."

"caturadhikam śatamaṣṭaguṇam dvāṣaṣṭistathā sahasrāṇām ayutadvayaviṣkambhasyāsanno vṛttapariṇāhaḥ."

"Translates to: Add four to 100, multiply by eight, and then add 62,000. By this rule the circumference of a circle with a diameter of 20,000 can be approached. Thus according to the rule ((4 + 100) × 8 + 62000)/20000 = 62832/20000 = 3.1416, which is accurate to five significant figures."

"tribhujasya phalashariram samadalakoti bhujardhasamvargah"

"Translates to: for a triangle, the result of a perpendicular with the half-side is the area."

"His value of π is a very close approximation to the modern value and the most accurate among those of the ancients. There are reasons to believe that he devised a particular method for finding this value. It is shown with sufficient grounds that he himself used it, and several later Indian mathematicians and even the Arabs adopted it. The conjecture that Aryabhata's value of π is of Greek origin is critically examined and is found to be without foundation. He discovered this value independently and also realised that π is an irrational number. He had the Indian background, no doubt, but excelled all his predecessors in evaluating π. Thus the credit of discovering this exact value of π may be ascribed to the celebrated mathematician, Aryabhata I."

"He is the master who, after reaching the furthest shores and plumbing the inmost depths of the sea of ultimate knowledge of mathematics, kinematics and spherics, handed over the three sciences to the learned world."

"... it is extremely likely that Aryabhata knew the sign for zero and the numerals of the place value system. This supposition is based on the following two facts: first, the invention of his alphabetical counting system would have been impossible without zero or the place-value system; secondly, he carries out calculations on square and cubic roots which are impossible if the numbers in question are not written according to the place-value system and zero."

"He gave more elegant rules for the sum of the squares and cubes of an initial segment of the positive integers. The sixth part of the product of three quantities consisting of the number of terms, the number of terms plus one, and twice the number of terms plus one is the sum of the squares. The square of the sum of the series is the sum of the cubes."

"His work, called Aryabhatiya, is composed of three parts, in only the first of which use is made of a special notation of numbers. It is an alphabetical system in which the twenty-five consonants represent 1-25, respectively; other letters stand for 30, 40, …., 100 etc. The other mathematical parts of Aryabhatiya consists of rules without examples. Another alphabetic system prevailed in Southern India, the numbers 1-19 being designated by consonants, etc."

"The greatest of Hindu astronomers and mathematicians, Aryabhata, discussed in verse such poetic subjects as quadratic equations, sines, and the value of π; he explained eclipses, solstices and equinoxes, announced the sphericity of the earth and its diurnal revolution on its axis, and wrote, in daring anticipation of Renaissance science: “The sphere of the stars is stationary, and the earth, by its revolution, produces the daily rising and setting of planets and stars.”"

"The Hindus were not so successful in geometry. In the measurement and construction of altars the priests formulated the Pythagorean theorem (by which the square of the hypotenuse of a right-angled triangle equals the sum of the squares of the other sides) several hundred years before the birth of Christ. Aryabhata, probably influenced by the Greeks, found the area of a triangle, a trapezium and a circle, and calculated the value of π (the relation of diameter to circumference in a circle) at 3.1416—a figure not equaled in accuracy until the days of Purbach (1423-61) in Europe. Bhaskara crudely anticipated the differential calculus, Aryabhata drew up a table of sines, and the Surya Siddhanta provided a system of trigonometry more advanced than anything known to the Greeks."

"Aryabhata is acknowledged as one of the astute astronomers of early India. His school of astronomy is well known and widespread all over India, especially in the South...Of late there is a tendency to spell his name as “Aryabhatta”. While Aryabhata himself mentions Kali 3600 to be the date of his composing the work, some say that Kali 3600 is the date of his birth. A view has been broached that Aryabhata hailed from Kerala."

"Of late, there has been a tendency to spell the name as “Aryabhatta” with the suffix “bhatta”. Two artificial satellites sent up into space by Indian scientists are given the names “Aryabhatta I” and “Aryabhatta II”. Some modern writers also make use of this spelling."

"He mentions his name at three places only as “Aryabhata”, towards the beginning and ending Verses of his work Aryabhatiya,"

"He gives a clue to his date of birth in his Aryabhatiya... The date works out to the end of the Kali year 3600, corresponding to the Saka year 421, the date being 21 March 499 ...and that he composed the Aryabhatiya when he was 25 years old, i.e. in Saka 444 or AD 522. Page 4"

"The second reason adduced, viz., that Aryabhata should have hailed from Kerala is fragile. Besides the Aryabhatan system being prevalent in this land,-“all” commentaries on Aryabhaflya have been produced by Kerala astronomers really does not stand scrutiny."

"He hailed from the Asmaka country, which comprised the present South Gujarat and North Maharashtra, through which the rivers Godavari and Narmada flowed. He flourished at Pataliputra (modern Patna) in the ancient Magadha country (now Bihar) where he composed his works, the 'Aryabhatiya and arya-siddhanta'."

"He was a celebrated astronomer and mathematician of the classical period of the Gupta Dynasty...He played an important role in shaping scientific astronomy in India. He is designated as Arayabhata I to differentiate him from Arya Bhata II who flourished much later (ca. AD 950-1100) and who wrote the Mahasiddhanta."

"His fame rests mainly on his Aryabhatiya, but from the writings of Varahamihira (Sixth century AD), Bhaskara I, and Brahmagupta (seventh century) it is clear that earlier he composed the Aryabhata Siddhantha (voluminous) is not extant. It is also called Ardharatrika Siddhanta, because in it the civil days were reckoned from one midnight to the next; 34 verses on astronomical instruments from this have been quoted by Ramakrishna Aradya."

"Aryabhatiya, an improved work, is product of mature intellect, which he wrote when he was 23 years old. Unlike in the Aryabhata siddhanta, the civil days are reckoned from one sunrise to the next, a practice which is still prevalent among the followers of Hindu calendar."

"The Aryabhatiya consists of four sections:1.Dasagitika (10+3 couplets in Giti meter); 2.Ganitapada (33 verses on mathematics); 3.Kala-kriyapada 25 verses on time-reckoning), and 4. Golapada (50 verses on spherical astronomy)"

"An Arabic translation of the Aryabhatiya entitled Zij-al-Arjabar (800 AD) is attributed to Ahwazi."

"Use of better planetary parameters, the innovations in astronomical methods, and the concise style of exposition of Aryabhatiya makes it an excellent text book on Astronomy. As opposed to the geostationary theory, Aryahabata held the view that the earth rotates on its axis. His estimate of the period of the sidereal rotation of earth was 23 hours 56 min, and 4.1 s is close to the actual value."

"He was the father of the Indian epicyclic astronomy which resulted in the planetary theory that determines more accurately the true positions and distances of the planets (including the Sun and the Moon)...was also the first to produce celestial latitudes...proposed the scientific cause of eclipses as against the mythological demon Rahu [Moon's node]. His ideas resulted in the new school of Indian Astronomy – the Āryabhata School Āryapakșa based on the text of Āryabhatīțya."

"The peculiar system of alphabetic numerals evolved by him with 33 consonants of the Sanskrit alphabet (Nagari script) denoted various numbers in conjunction with vowels which themselves did not represent any numerical value. For example khyughr (=khu+yu+ghr) is denoted by 2x100^2 +30x100^2+4x10^3 =4,300,000 which is the number of revolutions of the Sun in a yuga (epoch)"

"The development of Indian trigonometry, based on sine as against chord of the Greeks, a necessity for astronomical calculations with his own concise notation which expresses the full sine table in just one couplet for easy remembrance. One of the two methods suggested by him for the sine table is based on the property that the second order sine differences were proportional to sines themselves."

"In geometry his greatest achievement was an accurate value of π. His rule is stated as: dn^2+(2a-d)n=2s, which implies the approximation 3.1416 which is correct to the last decimal place."

"...he flourished in Kusumapura—near Patalipurta (Patna), then the capital of the Gupta dynasty — where he composed at least two works, Aryabhatiya (c. 499) and the now lost Aryabhatasiddhanta. Aryabhatasiddhanta circulated mainly in the northwest of India and, through the Sāsānian dynasty (224–651) of Iran, had a profound influence on the development of Islamic astronomy. Its contents are preserved to some extent in the works of Varahamihira (flourished c. 550), Bhaskara I (flourished c. 629), Brahmagupta (598–c. 665), and others. It is one of the earliest astronomical works to assign the start of each day to midnight."

"Aryabhatiya...written in verse couplets ...contains astronomical tables and Aryabhata’s system of phonemic number notation, the work is characteristically divided into three sections: Ganita (“Mathematics”), Kala-kriya (“Time Calculations”), and Gola (“Sphere”)."

"In Ganita, he names the first 10 decimal places and gives algorithms for obtaining square and cubic roots, utilizing the decimal number system. Then he treats geometric measurements — employing 62,832/20,000 (= 3.1416) for π—and develops properties of similar right-angled triangles and of two intersecting circles."

"With Kala-kriya he turned to astronomy — in particular, treating planetary motion along the ecliptic. The topics include definitions of various units of time, eccentric and epicyclic models of planetary motion (see Hipparchus for earlier Greek models), planetary longitude corrections for different terrestrial locations, and a theory of “lords of the hours and days” (an astrological concept used for determining propitious times for action)."

"...spherical astronomy in Gola, where he applied plane trigonometry to spherical geometry by projecting points and lines on the surface of a sphere onto appropriate planes. Topics include prediction of solar and lunar eclipses and an explicit statement that the apparent westward motion of the stars is due to the spherical Earth’s rotation about its axis. He also correctly ascribed the luminosity of the Moon and planets to reflected sunlight."

"Though its fame is much restricted by its specialized nature, there is no doubt that Panini's grammar is one of the greatest intellectual achievements of any ancient civilization, and the most detailed and scientific grammar composed before the 19th century in any part of the world."

"Ashtadhyayi Sanskrit treatise on grammar was written in the 6th to 5th century BCE by the Indian grammarian Panini. This work set the linguistic standards for Classical Sanskrit. It sums up in 4,000 sutras the science of phonetics and grammar that had evolved in the Vedic religion. Panini divided his work into eight chapters, each of which is further divided into quarter chapters, beyond defining the morphology and syntax of Sanskrit language."

"Ashtadhyayi distinguishes between usage in the spoken language and usage that is proper to the language of the sacred texts. The Ashtadhyayi is generative as well as descriptive. With its complex use of metarules, transformations, and recursions, the grammar in Ashtadhyayi has been likened to the Turing machine, an idealized mathematical model that reduces the logical structure of any computing device to its essentials."

"Panini, the ancient grammarian (probably belonged to 5th century or sixth century BC) mentions a character called Vasudeva son of Vasudeva, and also mentions Kaurava and Arjuna which testifies to Vasudeva Krishna, Arjuna and Kauravas being contemporaries. Megasthenes (350-290 BC), a Greek ethnographer and an ambassador of Seleucus I to the court of Chandragupta Maurya mentioned about Herakles in his famous work Indica. Many scholars have suggested that the deity identified as Herakles was Krishna"

"Grammars (vyākaraṇas) concern the description of speech forms (śabda) considered to be correct (sādhu) through derivation and thereby serve to make understood the usage found in the Vedas. The grammar that was granted the status of a Vedāṅga is that of Pāṇini. This work is referred to in toto as a śabdānuśāsana (means of instruction of correct speech forms); since the core of Pāṇini’s work comprises the eight chapters of sūtras that serve to describe both the current language of his time and features particular to Vedic, it also bears the name Aṣṭādhyāyī (“Collection of Eight Chapters”)."

"The accepted cultivated speech of the contemporary language that Pāṇini describes in his Aṣṭādhyāyī must have coexisted with more vernacular varieties of speech in which there were features belonging to the Middle Indo-Aryan division of the language group. Several facts support this view."

"The Pāṇinian commentator Kātyāyana (c. 3rd–4th century BCE) knew of the coexistence of Middle Indic forms with earlier ones. There is a Pāṇinian rule that provides that verb bases listed in an appendix to the Aṣṭādhyāyī have the class name dhātu (verbal base, root). Kātyāyana discusses whether one could define verbal bases semantically and thereby possibly do without the verb list. He remarks that even if one defines a verbal base as denoting an action, the roots must be listed in order to preclude the possibility that constituents of terms such as āṇapayati/āṇavayati ‘commands’ be assigned the class name in question; āṇapayati/āṇavayati is a Middle Indic counterpart of Sanskrit ājñāpayati."

"The current language Pāṇini describes is very close in structure to the late Vedic found in certain Brāhmaṇa texts. As noted earlier, scholars have recognized other varieties of Sanskrit. Epic Sanskrit is so called because it is represented principally in the two epics, Mahābhārata (“Great Epic of the Bhārata Dynasty”) and Rāmāyaṇa (“Romance of Rāma”). In the latter the term saṃskṛta ‘adorned, cultivated, purified (by grammar)’ is encountered, possibly for the first time with reference to the language. The date of composition for the core of early Epic Sanskrit is considered to be in the centuries just preceding the Common Era."

"Panini’s grammar is the earliest scientific grammar in the world, the earliest extant grammar of any language, and one of the greatest ever written. It was the discovery of Sanskrit by the West, at the end of the 18th century, and the study of Indian methods of analyzing language that revolutionized our study of language and grammar, and gave rise to our science of comparative philology … The study of language in India was much more objective and scientific than in Greece or Rome. The interest was in empirical investigation of language, rather than philosophical and syntactical. Indian study of language was as objective as the dissection of a body by an anatomist."

"The grammar of Panini stands supreme among the grammars of the world, alike for its precision of statement, and for its thorough analysis of the roots of the language and of the formative principles of words. By employing an algebraic terminology, it attains a sharp succinctness unrivalled in brevity, but at times enigmatical. It arranges, in logical harmony, the whole phenomena which the Sanskrit language presents, and stands forth as one of the most splendid achievements of human invention and industry. So elaborate is the structure, that doubts have arisen whether its complex rules of formation and phonetic change, its polysyllabic derivatives, its ten conjugations with their multiform aorists and long array of tenses, could ever have been the spoken language of a people."

"Sanskrit is constructed like geometry and follows a rigorous logic. It is theoretically possible to explain the meaning of the words according to the combined sense of the relative letters, syllables and roots. Sanskrit has no meanings by connotations and consequently does not age. Panini's language is in no way different from that of Hindu scholars conferring in Sanskrit today."

"The author of the oldest extant Sanskrit grammar was Panini, a native of extreme north-west India, ... His work consists of nearly 4000 aphorisms, each of which owing to the extreme conciseness of the style, generally consists of not more than two or three words. Hence, the whole grammar could be printed within the compass of about thirty-five octavo pages. Yet it describes the entire Sanskrit language with a completeness which has never been equalled elsewhere. Thus it is at once the shortest and the fullest grammar in the world."

"The most interesting non-Western grammatical tradition—and the most original and independent—is that of India, which dates back at least two and one-half millennia and which culminates with the grammar of Panini, of the 5th century BCE. There are three major ways in which the Sanskrit tradition has had an impact on modern linguistic scholarship."

"Roman alphabet: siṃho vyākaraṇasya kartur aharat prāṇān priyān pāṇineḥ"

"English translation: A lion took the dear life of Panini, author of the grammatical treatise."

"His teacher was Varsa, and he was a contemporary of Katyayana, Vyadi, and Indradatta. Panini is said to have secured the favour of Shiva and obtained from him the alphabet in the form of fourteen pratyadhara sutras."

"Pingala was the brother of Panini."

"There is in the rules or definitions (sutras) of Panini a remarkably subtle and penetrating account of Sanskrit grammar. The construction of sentences, compound nouns, and the like is explained through ordered rules operating on underlying structures in a manner strikingly similar in part to modes of modern theory. As might be imagined, this perceptive Indian grammatical work held great fascination for 20th-century theoretical linguists. A study of Indian logic in relation to Paninian grammar alongside AristotleAristotelian]] and Western logic in relation to Greek grammar and its successors could bring illuminating insights."

"The word `Sanskrit' means “prepared, pure, refined or prefect”. It was not for nothing that it was called the `devavani' (language of the Gods). It has an outstanding place in our culture and indeed was recognized as a language of rare sublimity by the whole world. Sanskrit was the language of our philosophers, our scientists, our mathematicians, our poets and playwrights, our grammarians, our jurists, etc. In grammar, Panini and Patanjali (authors of Ashtadhyayi and the Mahabhashya) have no equals in the world; in astronomy and mathematics the works of Aryabhata, Brahmagupta and Bhaskara opened up new frontiers for mankind, as did the works of Charaka and Sushruta in medicine."

"The issue of theism vis-à-vis atheism, in the ordinary senses of the English words, played an important role in Indian thought. The ancient Indian tradition, however, classified the classical systems (darshanas) into orthodox (astika) and unorthodox (nastika). Astika does not mean “theistic,” nor does nastika mean “atheistic.” Panini, a 5th-century-BCE grammarian, stated that the former is one who believes in a transcendent world (asti paralokah) and the latter is one who does not believe in it (nasti paralokah)."

"We pass at once into the magnificent edifice which bears the name of Panini as its architect and which justly commands the wonder and admiration of everyone who enters, and which, by the very fact of its sufficing for all the phenomenon which language presents, bespeaks at once the marvelous ingenuity of its inventor and his profound penetration of the entire material of the language."

"The grammar of Panini is one of the most remarkable literary works that the world has ever seen, and no other country can produce any grammatical system at all comparable to it, either for originality of plan or analytical subtlety."

"For example, the great linguist Panini gave the concept for meta-language-and constructed one-thousands of years before computer scientists began exploring the same idea. No one has been able to match him to this day."

"Classical Sanskrit theatre flourished during the first nine centuries CE. Aphorisms on acting appear in the writings of Panini, the Sanskrit grammarian of the 5th century BCE, and references to actors, dancers, mummers, theatrical companies, and academies are found in Kautilya’s book on statesmanship, the Artha-shastra (4th century BCE)."

"The medicine of Sushruta is considerably older than the ninth century; and the grammar of Panini probably precedes Christianity."

"Pāṇini had before him a list of irregularly formed words, which survives, in a somewhat modified form, as the Uṇādi Sūtra. There are also two appendixes to which Pāṇini refers: one is the Dhātupāṭha, "List of Verbal Roots," containing some 2000 roots, of which only about 800 have been found in Sanskrit literature, and from which about fifty Vedic verbs are omitted; the second is the Gaṇapāṭha, or "List of Word-Groups," to which certain rules apply. These gaṇas were metrically arranged in the Gaṇaratna-mahodadhi, composed by Vardhamāna in 1140 A.D."

"Among the earliest attempts to explain Pāṇini was the formulation of rules of interpretation or paribhāshās; a collection of these was made in the last century by Nāgojibhaṭṭa in his Paribhāshenduçekhara."

"Next we have the Vārttikas or "Notes" of Kātyāyana (probably third century B.C.) on 1245 of Pāṇini's rules, and, somewhat later, numerous grammatical Kārikās or comments in metrical form: all this critical work was collected by Patanjali in his Mahābhāshya or "Great Commentary," with supplementary comments of his own. He deals with 1713 rules of Pāṇini."

"The Mahābhāshya was commented on in the seventh century by Bhartṛihari in his Vākyapadīya which is concerned with the philosophy of grammar, and by Kaiyaṭa (probably thirteenth century). About 650 A.D. was composed the first complete commentary on Pāṇini, the Kāçikā Vṛitti or "Benares Commentary," by Jayāditya and Vāmana."

"In the fifteenth century Rāmachandra, in his Prakriyā-kaumudī, or "Moonlight of Method," endeavoured to make Pāṇini's grammar easier by a more practical arrangement of its matter. Bhaṭṭoji's Siddhānta-kaumudī (seventeenth century) has a similar aim; an abridgment of this work, the Laghu-kaumudī, by Varadarāja is commonly used as an introduction to the native system of grammar. Among non-Pāṇinean grammarians may be mentioned Chandra (about 600 A.D.), the pseudo-Çākaṭāyana (later than the Kāçikā), and, the most important, Hemachandra (12th century), author of a Prākrit grammar."

"Panini was a Sanskrit grammarian who gave a comprehensive and scientific theory of phonetics, phonology, and morphology. Sanskrit was the classical literary language of the Indian Hindus and Panini is considered the founder of the language and literature. It is interesting to note that the word "Sanskrit" means "complete" or "perfect" and it was thought of as the divine language, or language of the gods."

"A treatise called Astadhyayi (or Astaka) is Panini's major work. It consists of eight chapters, each subdivided into quarter chapters. In this work Panini distinguishes between the language of sacred texts and the usual language of communication. Panini gives formal production rules and definitions to describe Sanskrit grammar. Starting with about 1700 basic elements like nouns, verbs, vowels, consonants he put them into classes. The construction of sentences, compound nouns etc., is explained as ordered rules operating on underlying structures in a manner similar to modern theory. In many ways Panini's constructions are similar to the way that a mathematical function is defined today."

"Sanskrit is a scientific and systematic language. Its grammar is perfect and has attracted scholars worldwide. Sanskrit has a perfect grammar which has been explained to us by the world's greatest grammarian Panini."

"The most successful, hence most prominent amongst these grammarians was Panini. His grammar, surpassing all others in tightness and precision, became the standard and remained so undisputedly until today. Panini was able to joint the original devanagari language into an exact framework of rules, thus preserving it for the posterity. Since his time, this language is called Sanskrit, “joined together, refined”."

"Pāṇini's grammar defines Classical Sanskrit, so Pāṇini by definition lived at the end of the Vedic period. He notes a few special rules, marked chandasi ("in the hymns") to account for forms in the Vedic scriptures that had fallen out of use in the spoken language of his time. These indicate that Vedic Sanskrit was already archaic, but still a comprehensible dialect."

"It is not certain whether Pāṇini used writing for the composition of his work, though it is generally agreed that he knew of a form of writing, based on references to words such as "script" and "scribe" in his Ashtadhyayi. These must have referred to Aramaic or early Kharosthi writing. It is believed by some that a work of such complexity would have been difficult to compile without written notes, though others have argued that he might have composed it with the help of a group of students whose memories served him as 'notepads' (as is typical in Vedic learning). Writing first reappears in India in the form of the Brāhmī script from the 3rd century BC in the Ashokan inscriptions."

"While Pāṇini's work is purely grammatical and lexicographic, cultural and geographical inferences can be drawn from the vocabulary he uses in examples, and from his references to fellow grammarians, which show he was a northwestern person. New deities referred to in his work include Vasudeva (4.3.98). The concept of dharma is attested in his example sentence dharmam carati "he observes the law" (cf. Taittiriya Upanishad 1.11)."

"Pingala and Panini (fifth century BCE) along with the likes of Aryabhata, Bhaskara and Brahmagupta are the pillars of ancient Indian mathematics. Astonishingly, Panini’s immortal fame is not even as a mathematician but as the definitive Sanskrit grammarian. But he also “introduced abstract symbols to denote various subsets of letters and words that would be treated in some common way in some rules; and he produced rewrite rules that were to be applied recursively in a precise order”, notes Mumford, “one could say without exaggeration that he (Panini) anticipated the basic ideas of modern computer science”."

"At a very early date India began to trace the roots, history, relations and combinations of words. By the fourth century B.C. she had created for herself the science of grammar, and produced probably the greatest of all known grammarians, Panini. The studies of Panini, Patanjali (ca. 150 A.D.) and Bhartrihari (ca. 650) laid the foundations of philology; and that fascinating science of verbal genetics owed almost its life in modern times to the rediscovery of Sanskrit."

"Linguistics is an Indian science par excellence, and the entire modern discipline of modern linguistics in indebted to the Indian grammarians of the 1st millennium BCE. In India, it has briefly missed out on the recent innovation of comparison between seriously different languages (as opposed to dialectal differences, well-known among Indian scholars). But here too, Indians should take some pride in the official birth of IE linguistics in Kolkata 1786... Linguistics started in Takṣaśĭla university, where Pāṇini taught 26 (or so) centuries ago. Indo-European linguistics started at the feet of Brahmin informers in Kolkata (as shown on a freeze in Oxford showing William Jones learning from Hindu Pandits)."

"His formalism even served in the twentieth century as the basis for the first high-level programming languages, as ALGOL60, which also work on the basis of a fully specified system of rules. Virtually all programming languages are written in formalism that uses Pāṇini’s linguistic notion of grammar... Compared with Pāṇini, the other linguistics from antiquity appears to be from a different world. No other work from Chinese, Greek or Roman literature comes close to Pāṇini’s grammar in terms of complexity of precision.... Western linguistics continued to be dominated by the taxonomic study of words after Priscian, and this situation did not change until the later Middle ages."

"The coming of Grothendieck's school of Algebraic Geometry to India should be attributed primarily to the efforts of Seshadri."

"I have often pondered over the roles of knowledge or experience, on the one hand, and imagination or intuition, on the other, in the process of discovery. I believe that there is a certain fundamental conflict between the two, and knowledge, by advocating caution, tends to inhibit the flight of imagination. Therefore, a certain naiveté, unburdened by conventional wisdom, can sometimes be a positive asset."

"In mathematics we agree that clear thinking is very important, but fuzzy thinking is just as important."

"Another project which keeps the Bourbaki name alive is the Seminaire Nicolas Bourbaki, which is a series of seminars, about 12–20 per year, on contemporary mathematics started in 1948. It is considered an honour to be invited to give a seminary in this seminar in this series; the only Indian to figure in the seminar so far is Harish Chandra, who gave a talk in the 1957–58 series and, apparently, thus lost the chance of winning the Fields medal in 1958! (Siegel was the chairman of the Fields medal committee in the ICM 1958!)"

"Archaeologically, this period is still blank… There is no special Aryan pottery… no particular Aryan or Indo-Aryan technique is to be identified by the archaeologists even at the close of the second millennium."

"Clearly, then, as Kosambi said, There must have been a small but active settlement of Indian traders in Mesopotamia …” And yet, as the same author noted, “The reciprocal settlement seems to have been absent or less prominent in India.”"

"Eminent Historians makes for depressing reading. It leaves one wondering as to what must be stirring in the minds and souls of these ‘eminent historians’, to make them sink to such depths of intellectual and moral degradation as would place them in the company of Lysenko and Goebbels... their disloyalty to the nation and the culture that has sustained and nourished them, and without which they would be nothing. Unlike Indian scientists and technologists who are recognized everywhere, in the world of humanities, these ‘eminent historians’ are utter nonentities, little more than crooked reflections of colonial stereotypes."

"“This is not true, but it doesn’t matter. The great mathematician Ramanujan was a clerk in the Madras port, while Einstein himself was serving as a clerk in the Swiss patent office when he discovered Relativity. (…) The idea of objectivity is beyond such minds; status means everything.”"

"“Indian Marxists in particular are singularly touchy about the whole thing and hate to be reminded that their pet dogma of the non-indigenous origin of the Vedic Aryan civilization is an offshoot of the same race theories that gave rise to Nazism.”"

"“what the history establishment has done through the models it has proposed for both the ancient and the medieval periods is to exactly reverse the historical picture”."

"“fabricating astronomical data going back thousands of years calls for knowledge of Newton’s Law of Gravitation and the ability to solve differential equations.”"

"This observation is puzzling, to say the least, and it is not at all clear what any of it has to do with ancient history. The first part about class and caste is standard Marxist fare. But Thapar's foray into futurology, the prediction that an "Aryan nation" could emerge from the discovery that the "Aryans" are native to India, is irrelevant to the history of India. It is relevant, however, to modern politics. The dreaded "Aryan nation" . . . was a European invention. Are we to discard evidence and cling to the Aryan-invasion theory because of a perceived political threat . . . ? (Rajaram and Frawley 1995, 15)"

"Thapar's comments are portrayed as "vintage Marxist rhetoric," which has "gratuitously drag[ed] in the bogey of the 'Aryan nation' . . . [as] a blatant attempt aimed at divert- ing attention away from the real issue" (Rajaram 1993, 33)."

""English translations of the Rigveda . . . represent a massive misinterpretation built on the preconception that the Vedas are the primitive poetry of nomadic barbarians. Nothing could be further from the truth" (Rajaram 1995, xvi)."

"For all its monopoly of the Indian history establishment, this influential group calling itself Marxists has made no contribution to history. The central theme of their work is that Hindus have contributed nothing to Indian civilisation: Everything from the Vedas and Sanskrit to science and mathematics is a foreign import."

"Both of them (Witzel and Rajaram), probably very surprised to find each other in the same bed, assert that the Aryan debate is over and has been definitively decided. Both think that this debate only shows signs of life once in a while because of its political interest and in spite of its scholarly resolution. Only, Witzel thinks that the AIT has won the debate and its denial only survives because it is politically useful to the Hindutva forces, while Rajaram thinks the AIT has been refuted and only survives because it is politically useful to anti-Hindu forces as well as to various other political movements, including racism. It is this motive that he also discovers in Witzel... Rajaram presents Witzel as “more activist than scholar”, and lists as proofs his interventions to thwart Hindu proposals to eliminate the Aryan invasion theory from the chapter on Hindu history in California schoolbooks, and to ban Dr. Subramanian Swamy, after the latter’s anti-Muslim utterances, from teaching economics at Harvard. ... His scholarly contributions confine themselves to refuting the Aryan Invasion Theory, without proposing an alternative explanation for a linguistic kinship that he rejects. In this respect, his discovery of the relevance of the Seidenberg findings about the anteriority of Baudhayana’s mathematics to Babylonian mathematics (which dates Baudhayana’s late-Vedic writings dramatically earlier than hitherto assumed) remains pivotal in the Aryan debate. But for a presentation of the whole Aryan problem, he simply and willfully lacks the knowledge."

"Ironically, many of those expressing these anti-migrational views are emigrants themselves, engineers or technocrats like N. S. Rajaram... who ship their ideas to India from U.S. shores."

"The foundation of the Muslim League and Minto’s concessions had the effect of dividing the Hindus and Muslims into almost two hostile political camps. A remarkable example of this is afforded by a letter written about 1908 by Mr. Ziauddin Ahmad, later Vice- Chancellor of the Muslim University, Aligarh, to Mr. Abdulla Shuhrawardy, both of whom were then prosecuting their studies in Europe. Abdulla Shuhrawardy shared the national feelings which then characterized Indian students in Europe, and for this he was rebuked by Ziauddin in a letter from which we quote the following extract; “You know that we have a definite political policy at Aligarh, i.e. the policy of Sir Syed. I understand that Mr. Kirshna Varma has founded a society called ‘Indian Home Rule Society’ and: you are also one of its vice-presidents. Do you really believe that the Mohammedans will be profited if Home Rule be granted to India de lene. There is no doubt that this Home Rule is decidedly against the Aligarh policy...What I call the Aligarh policy is really the policy of all the Mohammedans generally—of the Mohammedans of Upper India particularly.” Mr. Asaf Ali wrote to Pandit Shyamji in September, 1909: “I am staying with some Muslim friends who do not like me to associate with nationalists; and, to save many unpleasant consequences, I do not want to irritate them unnecessarily.” Thus the Muslim antagonism to the Freedom Movement of India dates back to its beginning itself. (151ff)"

"One who was well versed in that science was called in ancient India as samkhyajna (the expert of numbers), parimanajna (the expert in measuring), sama-sutra-niranchaka (Uinform-rope-stretcher), Shulba-vid (the expert in Shulba) and Shulba-pariprcchaka (the inquirer into the Shulba). Of these term, viz, 'sama-sutra-niranchaka' perhaps deserves more particular notice. For we find an almost identical term, 'harpedonaptae' (rope-stretcher) appearing in the writings of the Greek Democritos (c. 440 BC). It seems to be an instance of Hindu influence on Greek geometry. For the idea in that Greek term is neither of the Greeks nor of their acknowledged teachers in the science of geometry, the Egyptians, but it is characteristically of Hindu origin." The English word 'Geometry' has a Greek root which itself is derived from the Sanskrit word 'Jyamiti'. In Sanskrit 'Jya' means an arc or curve and 'Miti' means correct perception or measurement."

"The sum of two positive quantities is positive; of two negative is negative; of a positive and a negative is their difference; or, if they are equal, zero. The sum of zero and negative is negative; of positive and zero is positive; of two zeros is zero (31)."

"In subtraction, the less is to be taken from the greater, positive from positive; negative from negative. When the greater, however, is subtracted from the less, the difference is reversed. Negative taken from zero becomes positive; and positive [taken from zero] becomes negative. Zero subtracted from negative is negative; from positive, is positive; from zero, is zero. When positive is to be subtracted from negative, and negative from positive, they must be thrown together (32-33)."

"The product of a negative quantity and a positive is negative; of two negatives, is positive; of two positives, is positive. The product of zero and negative, or of zero and positive, is zero; [the product] of two zeros, is zero. (34)."

"Positive, divided by positive, or negative by negative, is positive. Zero, divided by zero, is zero. Positive, divided by negative, is negative. Negative, divided by positive, is negative. Positive, or negative, divided by zero, is a fraction with that for denominator: or zero divided by negative or positive. (35-36)."

"The square of negative or positive is positive; of zero, is zero. The root of a square is such as was that from which it was raised [i.e. either positive or negative]. (37)."

"In a linear equation in one unknown, the difference of the two known terms taken in the reverse order, divided by the difference of the coefficients of the unknown (is the value of the unknown)."

"Of the unknowns, their squares, cubes, fourth powers, fifth powers, sixth powers, etc., addition and subtraction are performed of the like; of the unlike they mean simply their statement severally."

"Multiply half the difference of the tabular differences crossed over and to be crossed over by the residual arc and divide by 900’. By the result so obtained increase or decrease half the sum of the (two) differences, according as this (semi-sum) is less or greater than the difference to be crossed over. We get the true functional differences to be crossed over."

"The old calculations dealing with planets based on the system of Brahma have become erroneous in course of past ages and therefore I, the son of Jishnugupta would like to clarify them."

"Florian Cajori, the noted historian, summed up the matter in an extraordinarily suggestive manner: The perversity of fate has willed it that the equation y2 = nx2 + 1 should now be called Pell’s Problem, while in recognition of Brahmin scholarship it ought to be called the “Hindu Problem.” It is a problem that has exercised the highest faculties of some of our greatest modern analysts. Indian mathematical historians would like to call it the Brahmagupta–Bhaskara problem, keeping in mind that Bhaskar perfected Brahmagupta’s method of solution in the twelfth century; Bhaskara used “Chakravala”, or a cyclic process, to improve Brahmagupta’s method by doing away with the necessity of finding a trial solution."

"I adore that Brahman, also that science of calculation with the unknown, which is the one invisible root-cause of the visible and multiple-qualified universe, also of multitudes of rules of the science of calculation with the known."

"As out of Him is derived this entire universe, visible and endless, so out of algebra follows the whole of arithmetic with its endless varieties (of rules). Therefore, I always make obeisance to Shiva and also to (avyakta) ganita (algebra)."

"Let ‘so much as’ (yavattavat) be put for the value of the unknown quantity, and doing with that precisely what is proposed in the instance [i.e. what is proposed in the following], let two equal sides be carefully completed, adding or subtracting, multiplying or dividing, [as the case may require]. Subtract the unknown quantity of one side from that of the other; and the known number of the one from that of the other side. Then divide the remainder of the known quantity by the [coefficient of the] remaining unknown: the quotient is the distinct value of the unknown quantity."

"A person has 300 rupees and 6 horses; and another person has 10 horses and 100 rupees debt; and the property of the two is equal; and the price of the horses is the same; what then is the value of each?"

"On a plane surface describe a circle of any specified radius with a pair of compasses. Mark on its circumference 360 degrees. Draw the east-to-west and north-to-south lines through its center. These lines will divide the circle into quadrants, which should be taken into consideration in the leftwise manner (i.e. anti-clockwise)."

"The Rsines of any two arcs of a circle are reciprocally multiplied by their Rcosines; the products are then divided by the radius; the sum of the quotients is equal to the Rsine of the sum of the two arcs, and their difference is the Rsine of the difference of the arcs."

"Without the knowledge of upapattis, by merely mastering the calculations (ganita) described here, from the madhyamadhikara (the first chapter of Siddhantashiromani) onwards, of the [motion of the] heavenly bodies, a mathematician will not be respected in the scholarly assemblies; without the upapattis he himself will not be free of doubt…."

"The Moon, moving like a cloud in a lower sphere, overtakes the Sun [and obscures its shining disk by its own dark body] hence it arises that the western side of the Sun’s disk is first obscured, and that the eastern side is the last part relieved from the Moon’s dark body: and to some places the Sun is eclipsed and to others is not eclipsed (although he is above the horizon) on account of their different orbits. (1)At the change of the Moon it often so happens that an observer placed at the center of the earth, would find the sun when far from the zenith, obscured by the intervening body of the moon, whilst another observer on the surface of the earth will not at the same time find him to be so obscured, as the moon will appear to him [on the higher elevation] to be depressed from the line of vision extending from his eye to the sun. Hence arises the necessity for the correction of parallax in celestial longitude and parallax in latitude in solar eclipses in consequence of the difference of the distances of the sun and the moon. (2) When the sun and moon are in opposition, the earth’s shadow envelopes the moon in darkness. As the moon is actually enveloped in darkness, and as the earth’s shadow and the moon which enters it, are at the same distance from the earth, there is therefore no call for the correction of the parallax in a lunar eclipse. (3)"

"As the moon moving eastward enters the dark shadow of the earth: therefore its eastern side is first of all involved in obscurity, and its western is the last portion of its disc which emerges from darkness as it advances in its course. (4) As the sun is a body of vast size, and the earth insignificantly small in comparison: the shadow made by the sun from the earth is therefore of a conical form terminating in a sharp point. It extends to a distance considerably beyond that of the moon’s orbit. (5)"

"The length of the earth’s shadow, and its breadth at the part traversed by the moon, may be easily found by proportion. In the lunar eclipse the earth’s shadow is northwards or southwards of the moon when its latitude is south or north. Hence the latitude of the moon is here to be supposed inverse (i.e. it is to be marked reversely in the projection to find the center of the earth’s shadow from the moon.) (6)"

"If the earth were supported by any material substance or living creature, then that would require a second supporter, and for that second a third would be required. Here we have the absurdity of an interminable series. If the last of the series be supposed to remain firm by its own inherent power, then why may not the same power be supposed to exist in the first, that is in the Earth? For is not the Earth one of the forms of the eight-fold divinity i.e. of Shiva? (Goladhyaya, III.4)"

"The property of attraction is inherent in the Earth. By this property the Earth attracts any unsupported heavy thing towards it: The thing appears to be falling [but it is in a state of being drawn to the Earth]. The ethereal expanse being equally outspread all around, where can the Earth fall? (Goladhyaya III.6)"

"As the one-hundredth part of the circumference of a circle is (scarcely different from) a plane, and as the Earth is an excessively large body, and a man exceedingly small (in comparison), the whole visible portion of the Earth consequently appears to a man on its surface to be perfectly plane."

"From a bunch of lotuses, one third is offered to Lord Shiva, one fifth to Lord Vishnu, one sixth to the sun, one fourth to the goddess. The remaining six are offered to the Guru. Find quickly the number of lotuses in the bunch."

"Arjuna became furious in the war and in order to kill Karna, picked up some arrows. With half the arrows, he destroyed all of Karna’s arrows. He killed all of Karna’s horses with four times the square root of the arrows. He destroyed the spear with six arrows. He used one arrow each to destroy the top of the chariot, the flag, and the bow of Karna. Finally he cut off Karna’s head with another arrow. How many arrows did Arjuna discharge?"

"There is no change in infinite (khahara) figure if something is added to or subtracted from it. It is like: there is no change in the infinite Lord Vishnu due to the dissolution or creation of abounding living beings."

"In a triangle or a polygon, it is impossible for one side to be greater than the sum of the other sides. It is daring for anyone to say that such a thing is possible. If an idiot says that there is a quadrilateral of sides 2, 6, 3, 12 or a triangle with sides 3, 6, 9, explain to him that they don’t exist."

"The earth attracts inert bodies in space towards itself. The attracted body appears to fall down on the earth. Since the space is homogeneous, where will the earth fall?"

"On the subject of demonstrations, it is to be remarked that the Hindu mathematicians proved propositions both algebraically and geometrically: as is particularly noticed by Bhaskara himself, towards the close of his algebra, where he gives both modes of proof of a remarkable method for the solution of indeterminate problems, which involve a factum of two unknown quantities."

"Bhaskara […] does not pretend himself to be the inventor, he assumes no character but that of a compiler."

"Almost any trouble and expense would be compensated by the possession of the three copious treatises on algebra from which Bhaskara declares he extracted his Bijaganita, and which in this part of India are supposed to be entirely lost."

"Reuben Burrow […] says, he was told by a pundit, that some time ago there were other treatises of algebra."

"This method is supreme above all praise; it is certainly the finest thing accomplished in number theory before Lagrange."

"A part of a circle is of the form of a bow, so it is called the ‘bow’ (dhanu). The straight line joining its two extremities is the ‘bow-string’ (jiva). It is really the ‘full-chord’ (samasta-jya). Half of it is here (called) the ‘half-chord’ (ardha-jya), and half that arc is called the ‘bow’ of that half-chord. In fact the Rsine (jya) and Rcosine (kotijya) of that bow are always half chords. [24]"

"The radius into the sine divided by the cosine is the first quote: this multiplied by the square of the sine, and divided by the square of the cosine, is the second quote; this second, and those obtained continually in the same way, multiply and divide by the square of the sine and the square of the cosine respectively: divide the quotes in order by 1, 3, 5, 7,11, etc. respectively, and the difference of the sum of the first, third, fifth, etc. and of the second, fourth, sixth, etc., will be the arc whose sine was taken."

"Square the diameter and multiply the product by 12, and extract the root of this product; the root obtained will be the modulus of odd quotes, which if you divide by 3, the quotient will be the modulus of even quotes. Divide each modulus continually by 9, and the quotient thus obtained from the former, divide by double the numbers 1, 3, 5, 7, 9, etc. minus 1 respectively, and the quotient obtained by the latter, by double the number 2, 4, 6, 8, 10, etc. minus 1 respectively, add up the new obtained quotes, and subtract the sum of those gotten from the even from the sum of those gotten from the odd modulus, the remainder is the circumference of the circle. Square the diameter and multiply the product by 12, and extract the root of this product; this root divide continually by 3, and the quotients thus obtained by 1, 3, 5, 7, 9, 11, etc., and subtract the sum of the second, fourth, sixth, eighth of the last obtained quotes from the sum of the first, third, fifth, seventh, ninth, etc. If you do thus, and measure the diameter of a great circle by 100000000000000000 equal parts, the circumference will be equal to 314159265358979324 of such parts."

"The diameter multiplied by four and divided by unity (is found and saved). Again the products of the diameter and four are divided by the odd numbers like three, five, etc., and the results are subtracted and added in order."

"Why is it that the actual value is left out and this very near value stated? Let me say. It is impossible to state the actual value. Why? That unit which leaves no remainder when the diameter is measured will leave a remainder if used again for measuring circumference. Likewise, the unit which leaves no remainder in the measure of the circumference will leave a remainder in the diameter if measured by the same unit. Hence if both (the diameter and circumference) are measured by the same unit, a remainderless state is never attained. Even if this is carried out farther to a great extent only diminution of the remainder can be obtained but absence of remainder can never be obtained— this is the meaning."

"One has to realize that the five siddhantas [i.e. astronomical systems] had been correct at a particular time. Therefore, one should search for a siddhanta that does not show discord with actual observations (at the present time). Such accordance with observation has to be ascertained by (astronomical) observers during times of eclipses etc. When siddhantas show discord, that is, when an earlier siddhanta is in discord, observations should be made of revolutions etc. (which would give results in accord with actual observations) and a new siddhanta enunciated."

"One has to accept that [each of ] the five siddhantas had been authoritative at one time [though they might not be so now]. Therefore one has to look for a system which tallies with observation. The said tallying has to be verified by contemporary experimenters at the time of eclipses etc."

"[Madhava] took the decisive step onwards from the finite procedures of ancient mathematics to treat their limit-passage to infinity which is the kernel of modern classical analysis."

"The sun moves towards south or north along the ecliptic every moment. Therefore, the direction [determined by the Indian circle method] appears to be incorrect. The corrected direction will be [obtained by applying a correction] further by [using] the R.Sine of the declination.The difference of the R. Sine of the sun’s declination at the time of the shadow’s entry and exit [to and from the level circle] is multiplied by the hypotenuse [of the shadow] and divided by the R.cosine of the terrestrial latitude. The result is [the correction in terms of ] angulas etc. One should shift the western mark to the opposite direction to the sun’s course (ayana). Otherwise, one should shift the eastern mark to the same direction of the sun’s course. [Thus] the correct east west line is [obtained]."

"In writing about physics, as distinct from mathematics or astronomy, in early Indian traditions, one is immediately struck by the apparent paucity of material—the available commentaries in English suggest that there is little beyond the Purusa Sukta, the pancabhutis and atomism."

"It seems part of human nature that if one desires something strongly one pretends that it is true. If the pretence is carried out long enough, it becomes difficult to distinguish between pretence and reality."

"It is a common error to confound quasi-cyclic time with eternal recurrence. It was not generally believed that these cosmic cycles were exact or eternal. The whole possibility of deliverance – moksa, nirvāna – was premised on the idea that these cycles were neither exact nor eternal. (However, the category of cyclic time encourages such an error by suggesting that various types of cyclic time are the same.) In India, this was the traditional view of time and life after death held from before the time of the Buddha. The Lokāyata denied the belief in life after death as a fraud. An interesting feature of this denial is how Pāyāsi sought to establish the non-existence of the soul by performing some 37 experiments with dying men, and condemned felons. It is unlikely that such experiments were ever performed anywhere else."

"Moving to pragmatic and people-oriented standards rather than the Westerm-oriented standards of the elite will hopefully also restore the idea of science as relating to our immediate surroundings, both social and natural."

"The trigonometric values published by Clavius ... provide further circumstantial evidence that the Jesuits had obtained the latest Indian texts on mathematics and astronomy.’"

"the term “sine” derives from sinus meaning fold, from the Arabic jaib, meaning fold for a pocket. This was written as “jb” omitting the vowels, but was intended to be read as jı̄bā, from the Indian term jı̄vā corresponding to the earlier Sanskrit jyā used for the chord. Possibly, the name “Euclid” was inspired by a similar translation error made at Toledo regarding the term uclides which has been rendered by some Arabic authors as ucli (key) + des (direction, space). So, uclides, meaning “the key to geometry”, was possibly misinterpreted as a Greek name Euclides."

"The rope (or string) is flexible in more ways than one and can be used to do everything that can be done with a compass-box. It can further be used to measure the length of a curved line, impossible with the instruments in a compass- box. This is helpful for the measurement of angles, and the subsequent transition to trigonometry and calculus."

"The Elements not only acquired a theologically-correct origin, it also acquired a theologically-correct interpretation. Plato and Neoplatonists had linked geometry and mathematics to the soul. The revised interpretation rejected this linkage as heretical. Mathematics was reinterpreted as “a universal means of compelling argument”."

"We have seen a number of difficulties raised by sceptics about the belief in life after death; these difficulties evaporate in the context of cosmic recurrence."

"No Western historian, to my knowledge, has commented on the curious fact that the theory of planetary motion in the West developed without the availability of appropriate planetary data. To begin with, every purported observation in “Ptolemy’s” Almagest is fabricated, and obtained by back-calculation. There is not a single known exception to this."

"The history of astronomy and physics in texts should be fundamentally revised. It should be pointed out, for example, that a scientific evaluation of the evidence indicates that Claudius Ptolemy did not exist (this would also teach students a lesson on how and why to do physics practicals in a more genuine way). It should also point out that Copernicus was no revolutionary, that Newton was a deeply religious person, and that Einstein might have played legalistic tricks which a patent clerk is expected to know. There are many other aspects of history and physics nomenclature which need to be revised (in texts)."

"To recapitulate, in mathematics, the East-West civilizational clash may be represented by the question of pramâna vs proof: is pramâna (validation), which involves pratyaksa (the empirically manifest), not valid proof? The pratyaksa or the empirically manifest is the one pramâna that is accepted by all major Indian schools of thought, and this is incorporated into the Indian way of doing mathematics, while the same pratyaksa, since it concerns the empirical, is regarded as contingent, and is entirely rejected in Western mathematics. Does mathematics relate to calculation, or is it primarily concerned with proving theorems? Does the Western idea of mathematical proof capture the notions of ‘certainty’ or ‘necessity’ in some sense? Should mathematics-as-calculation be taught primarily for its practical value, or should mathematics-as-proof be taught as a spiritual exercise?"

"If one excludes the philosophy of science from the ambit of a study of its history, then one is obliged to do history with the default philosophy of science. In our case this means that one must then accept the present-day Western philosophy of mathematics, not only as a privileged philosophy, but as the only possible philosophy of mathematics."

"The second consequence follows from the first: for if the Indian infinite series were established using a method of calculation and demonstration that does not constitute a formal mathematical proof, valid according to the present-day belief in the potency of formalism, then the Indian infinite series may forever have to be consigned to the status of "proto- calculus", or at best "pre-calculus", for that is how Western historians of science would surely like to classify them, if at all they are compelled to link these Indian infinite series to the infinitesimal calculus in Europe. After all, Indian infinite series were very similar to, if not identical with, the series used by Cavalieri, Fermat, Pascal, Barrow, Gregory, and Wallis, and these efforts are already classified as “pre-calculus” by Western historians of science. While such a strategy of classification and labelling may suit the political interests and the morbid narcissism of the West, it works against the grain of history regarded as an attempt to reconstruct the past."

"This book, since it presents a new account of Indian history, inevitably involves a critique of Western history. However, some Western scholars, recognizing the intrinsic weakness of that history, tend to respond to any critique of Western history not by examining the evidence (which would expose it) but by launching personal attacks on the critic with labels—in this case, the label "Hindu nationalist" seems to commonly arise to the tongues of shallow scholars. Now I completely fail to see why the only choice one has is between different kinds of hate politics— why the rejection of Western racist history necessarily implies the acceptance of some other kind of hate politics. ... It is easy to find many people who oppose one kind of hate politics while being "soft" on another set: however, as stated above, I fail to see why one's choice should be restricted to different brands of hate politics. I am not in any such camp, my stated system of ethics does not admit hate politics of any kind, and I oppose all attempts to mix religion with politics... Suppose “Hindu nationalists” were to seize power, strangle dissent by passing laws to kill dissenters, in painful ways, and then continuously expand their power through multiple genocide for the next 1700 years. What sort of history would emerge? We do not need to imagine very hard, for we have a concrete model before us, in the sort of Western history that has been written since Eusebius! Because of the long history of brutal suppression of dissent in the West, various fantasies, contrary to the barest common sense, have been allowed to pile up, and these continue today to masquerade as the scholarly truth."

"Only when it started emerging from the Dark Age did Europe first come to know of the Elements—through 12th c. translations from Arabic into Latin by Adelard of Bath and Gerard of Cremona—after the capture of the Toledo library, and the setting up there of a translation factory. However, at this time of the Crusades, there was a strong sense of shame in learning from the Islamic enemy. Also at the time of the Inquisition, the fears that Toledo was a Trojan horse that would spread heresy could not be lightly discounted. The shame was contained by the strategy of "Hellenization"—all the world knowledge, up to the 11th c. CE found in the Arabic books (including, for example, Indian knowledge) was indiscriminately assigned an early Greek origin, with the Arabs assigned the role of mere transmitters (and the Indians nowhere in the picture). The fear of heresy was contained by the strategy of Christianization of this incoming knowledge, by reinterpreting it to bring it in line with the requirements of Christian theology."

"From the historiographic angle, the confounding of Euclid of Megara with Euclid the supposed author of the Elements is interesting. While the occurrence of such a mistake is understandable, its persistence for five centuries is not. The persistence of this error for centuries shows that that stories about "Euclid" were propagated, by historians in Europe, exactly in the uncritical manner of myth."

"Of course, it is well known from the philosophy of science that any evidence whatsoever can be made consistent with any theory whatsoever by introducing enough auxiliary hypotheses."

"There is not the slightest doubt that every piece of empirical evidence can be explained away by one who wants to hang on to the myth of Euclid, just as every piece of evidence against astrology can be explained away by those who make a living from it."

"Such forgeries were common enough... So unenviable was the reputation that priests had acquired in this matter that Isaac Newton spent 50 years of his life trying to undo the forgeries that he thought various priests had incorporated into the Bible, to serve their temporal ends. And the only answer to his scholarly and voluminous accusations was to hide them for some 250 years—in fact they still remain secret."

"note here that the long-standing claims of Euclid's existence, and the surprisingly flimsy evidence on which they are based, also provide an example of the de facto standards of evidence in historiography—standards to decide origin and transmission that should either be uniformly applied elsewhere or rejected here as well."

"The khichdi geometry in the NCERT text for Class 9 is indigestible because it has mixed up the Elements by mixing up elements that ought not to be taken together—like diazepam and alcohol—unless the object is to induce a comatose state."

"It would be rather pointless and confusing to retain in these books information that was incorrect or defective or inaccurate. That is to say, books on science and mathematics would naturally be propagated accretively, with the addition of numerous anonymous updates, though no one maintained a revision history. Certainly Arab authors in Baghdad, for example, were actively disinterested in verbatim translations, but were interested rather more in useful paraphrases and creative reworking."

"Nevertheless, this laughable hypothesis is exactly what has been adopted with the 12th and 16th c. sources of “Greek” or “Hellenic” tradition." Hence, virtually all the knowledge prevalent in the 11th c. world, as known to Indians and Arabs, is attributed to Greeks like Aristotle, Archimedes, and Ptolemy. The fact is that the knowledge in these 11th c. texts accurately reflects the knowledge that then prevailed—as is naturally to be expected. However, Western historians explain this fact not by the simple and natural hypothesis of accretive up- dating of the texts, but by the extraordinary claim that all (or most of) the contemporary knowledge of the 11th c. world was derived by transmission from the Greeks, who had anticipated these developments. There is no other, or direct, evidence that these Greek authors wrote anything at all. Thus, by way of evidence, this extraordinary‘theory of transmission simply begs the question! To complete the story, it is thought enough to supplement it with a speculative chronology, attached to Greek names, based on stray remarks of doubtful authenticity in late texts. This sort of story-telling may be perfectly consonant with the standards of theology (and most early Western historians were priests), but is completely unconvincing from a somewhat more sceptical and down-to-earth point of view."

"Attributing a book to a famous early source added not only to-the authority of the book, but also to its market price in what was evidently a flourishing book bazaar in Baghdad. That many books were fakes and falsely attributed to famous early sources is evident from the Eihrist of al Nadim, a Baghdad shopkeeper of the 10th c., who hence prepared this fihrist or list of books he regarded as genuine. Of course, al Nadim was a shopkeeper, not a scholar, and his concerns about genuineness were limited to saleability—so, common hearsay was good enough for him— and he is unlikely to have been bothered by a well-established fake."

"“Ptolemy’s” Almagest begins (as natural for an 11th c. text) with what look like paraphrases of controversies from the history of Indian astronomy, “Aristotle’s” syllogisms are remarkably similar to the Nyàya theory of syllogisms, "Aristotle" uses theories like those of "action by contact" and the same words like "aether" (= sky = dkdsa) long used in India, and his physics is as similar to Arabic physics as "Archimedes" is to 11th c. Arabic mathematics."

"So, in practice, Western history has used two standards of evidence for transmission: one ultra-lax standard of evidence for transmission from "Greeks", and another ultra-strict standard for transmission to the West. For cases of alleged transmission from the Greeks, mere speculations—a speculative chronology combined with speculative attribution—are regarded as ample evidence of transmission. In the other direction, similarity with a real earlier work, by a real author, together with a clear channel of transmission, do not prove anything, for there is always the possibility of repeated miracles by which any number of people in the West may independently reinvent things just when they could be transmitted."

"The Doctrine of Christian Discovery, which instigated the subsequent triple genocide in three continents of South and North America and later Australia—the only known successful cases of genocide in a literal sense—was explicitly proclaimed in papal bulls (Romanus Pontifex, 1454, and Inter Caetera 1493), which declared it the religious duty of Christians to kill and enslave all non-Christians. The first-hand descriptions of the genocide in the Americas provided by Las Casas (who accompanied Columbus) clearly show that it was religiously motivated, and that those engaged in the genocide thought they were doing their Christian duty by eliminating non-Christians and carrying out God’s will here on earth as it would be in hell."

"Therefore, it is hardly a matter of surprise that there is much similarity between Indian knowledge, and knowledge that has been attributed to the early Greeks based on late Arabic texts: for example, the astronomical model attributed to “Ptolemy” is remarkably similar to Indian astronomical models, "Aristotle's" theory of action by contact, using aether (sky —àkàsa) is as similar to the Nyaya theory as his syllogisms are to Nyàya syllogisms, etc."

"In the many centuries, since Toledo, that Western historians have been talking of transmission from the Greeks, who ever produced a Sanskrit manuscript of Ptolemy? Who ever proved that Aryabhata had seen such a Sanskrit manuscript? Yet every Western reference work on the subject asserts that Indian astronomy is transmitted from the Greeks. So is it the case that these reference works are all out of date, and that the standard of evidence for transmission has now changed? Does Owen Gingerich now deny transmission from the Greeks on the grounds that there is no evidence? Not at all; in the very same article he sticks to the entire fairy tale about transmission from the Greeks. So, it is not so much that the standards of evidence have changed, but that there are (even as of today) two simultaneous standards of evidence for transmission. One for transmission to the West, and another for purported transmission from the West. Not only is the judge biased, the very rules of evidence are biased!"

"So, similarity and precedence do not always establish transmission. Whether or not they establish transmission depends upon the direction of transfer. Thus, in practice, there are two standards of evidence for transmission: an ultra-lax standard for transmission from Greeks, and an ultra-strict standard for transmission to the West."

"In support of the West’s physical claim to the whole world, the Western history of science sought to establish an intellectual claim to all knowledge in the world, especially all scientific knowledge. To situate this claim in its proper perspective, we need to probe a little deeper to understand a bit of the unstated logic behind colonialism. According to the religious beliefs of the colonialists, such an intellectual claim of discovery, in turn, established the colonialist’s moral claim to the whole world. It was these “moral” claims that distinguished colonialism from a simple project of robbing the world by physical force."

"But the mysterious source of Mercator's precise trigonometric values, and his technique, remains unknown to this day. Mercator, who worked with Gemma Frisius at the Catholic University of Louvain, obviously had privileged access to information brought in by sailors and priests returning from India and China, via Antwerp. So it is hardly surprising that the "Mercator" projection is identical with a projection used in maps of the celestial globe from China from at least five centuries earlier—and the same principle could obviously be applied to the terrestrial globe. How- ever, since Mercator was arrested by the Inquisition, and was lucky to escape with his life, it is also not surprising that he kept his "pagan" sources of information a closely guarded secret. The tables of trigonometric values published by Clavius, in 1608, used the Indian de- finition of sines and cosines, and the then common Indian value for the radius of the circle. Hence, these tables far exceeded in accuracy the "tables of secants" provided by earlier nav- igational theorists like Stevin for calculation of loxodromes, which were (at the accuracy of) Aryabhata's values, known to the Arabs. It is hard to see how such accuracy (unprecedented for Europe) could even have been attempted without calculus techniques. Clavius, who au- thored the calendar reform proclaimed by pope Gregory, certainly had access to every bit of information brought in by the Jesuits, but could hardly be expected to be truthful enough to acknowledge his “pagan” sources. Since Clavius’ tables were published several years be- fore the first hint of the calculus “officially” appeared in Europe in the works of Kepler, and since Clavius provides no explanation of his method, it remains a mystery how these high- precision trigonometric values were calculated. The only reasonable explanation is that like his contemporaries, Tycho Brahe, who merely articulates Nilakantha’s astronomical model, or Scaliger, whose “Julian” day number system copies the Indian ahargana system, Clavius obtained his trigonometric values from India."

"We have seen that calculation of loxodromes involved the solution of a problem equivalent to the fundamental theorem of calculus. But that theorem was unknown to Europeans in the 16th c. How, then, did Mercator draw the chart? The abiding nature of the Mercator mystery is due to the fact that it cannot be appropriately solved within the framework of the Western historical narrative about the calculus. The mystery can be resolved by changing that narrative. It is hard to believe that Mercator drew his chart through sheer skill in draftsmanship. It is rather more likely that he had access to information from India or China, which he kept a secret. That this information was adequate to enable the calculation of loxodromes is evident from the fact that loxodromes were earlier used to map the zodiac, and a Chinese [Dunhuang] star map from ca. 950 follows the very same principle of isogonal cylindrical projection that has come to be known as the “Mercator” projection. This chart is reproduced in Needham’s volume."

"The trigonometric values published by Clavius, who was at the centre of the Jesuit web, provide further circumstantial evidence that the Jesuits had obtained the latest Indian texts on mathematics and astronomy, and had studied them. Thus, Clavius’ trigonometric values use exactly the Indian definition of the sine and also the same value of the radius?? used by Indian sources in stating Madhava’s sine values. Further, Clavius was unable to give any explanation for the way those trigonometric values were derived, and, obviously enough, the derivation of such precise values required essentially calculus techniques. Had Clavius himself discovered a striking new procedure, by which to obtain more precise trigonometric values, would he not have announced it, to establish his priority, especially since this was towards the end of his life? In fact, Clavius, though he published sophisticated trigonometric tables in his name, lacked a proper understanding of even elementary trigonometry, since he was unable to use trigonometry to determine a key navigational parameter—the size of the globe."

"When Indian astronomy works, translated by Jesuits in Cochin, started arriving in Europe, Tycho, as one of the most famous astronomers of his day, and the Mathematician of the Holy Roman Empire, would naturally have been chosen as the person to whom they were referred. Nilakantha's model was what later came to be called the “Tychonic” model, which Tycho was trying to check against observations. Why, after all, was Tycho so secretive about his papers, not even allowing his trusted assistant Kepler to see them? In any case, on Tycho's sudden death, Kepler obtained not just Tycho's observations, but also the rest of his papers which contained the underlying theory."

"Galileo's access to Jesuit sources at the Collegio Romano is well documented. Galileo did not himself take up the calculus because he did not quite understand it, as is clear from the difficulties and the various paradoxes of the infinite that he raised in his letters to Cavalieri. Thus, this state of affairs is better explained by supposing that there was a common body of Indian work related to the calculus, known to both Galileo and Cavalieri, and that Galileo was not satisfied with Cavalieri's interpretation of it, and not willing to risk his reputation, while Cavalieri was. Nevertheless, out of deference for his teacher, he waited five years before staking his claim."

"The influence of Cavalieri’s work on Torricelli and Roberval is well known. Roberval was a member of Mersenne’s discussion group, and was involved, along with Fermat and Pascal, in debating with Descartes, the validity of these new methods. There is a clear chain of influence from Cavalieri to Torricelli, to Wallis to Gregory and Newton. As is well known, while Newton acknowledged the influence of Wallis, Leibniz acknowledged the influence of Pascal on their respective works relating to the calculus. A diffusionist model for the calculus in Europe is, therefore, rather more appropriate than the simplistic Eurocentric model which gives all credit to Newton and/or Leibniz just because the two had a nasty priority dispute! There is further circumstantial evidence of transmission. The calculus methods of Cavalieri, Roberval, Fermat and Pascal are very similar to those of the Yuktsbhasd, TantrasangrahaVyakhya, Kriyakramakari."

"The issue of transmission does not end with the receipt of the calculus in Europe. Because of the epistemological differences between Indian and European mathematics, actual assimilation of the calculus took a long time. It is worthwhile trying to understand this assimilation process, since this sheds light on the historical as well as the contemporary mathematical situation, and since such a task seems never before to have been attempted by historians of mathematics, who have not acknowledged or understood the historical existence of epistemological differences within mathematics."

"Briefly, Europe inherited not one but two mathematical traditions: (i) from Greece and Egypt a mathematics that was spiritual, anti-empirical, proof-oriented, and explicitly religious, and (ii) from India via Arabs a mathematics that was pro-empirical, and calculation-oriented, with practical objectives.' Much mathematics taught at the K-12 level is of Indo-Arabic origin: (1) arithmetic, (2) algebra, (3) trigonometry, and (4) calculus. Despite the obviously different philosophical orientations of these two streams of mathematics Europe recognized only a single possible philosophy of a "universal" European mathematics, into which it forcibly sought to fit both mathematical streams."

"... formal mathematics is no more than a culturally-dependent system of aesthetics, ... it may continue to be taught like Western music..."

"The linkage of time perceptions to ethics applies also to Buddhism. The relevant notion of time here is the notion of paticca samuppada , an understanding of which was equated by the Buddha with an understanding of the dhamma. This is a deep and tricky point about Buddhist ethics"

"Unfortunately, there are double standards in the matter: one standard for Greek history, another for Indian."

"Witzel’s way of arguing, by concocting a false position for the opponent and attacking it, is unethical, whether it was done deliberately or because of lack of understanding."

"We have no truthful account of what happened next, for the written accounts that have come down to us are all from the viewpoint of the Christian priests. This is rather like describing a rape and murder from the viewpoint of the rapist and the murderer on the grounds that there is no other reliable source of evidence."

"Students need to be taught that belief in Einstein or Stephen Hawking is not less superstitious than the belief in Sai Baba, just because those figures are endowed with high scientific authority in the West. They should be taught to use right means of validating knowledge, without relying on authority. (This applies also to decision makers who should not rely on the privately expressed opinions of “experts”, since this may involve a conflict of interests, but should use public discussions.)"

"People have been indoctrinated to believe that any attempt to correct Western history is necessarily chauvinistic. This latter belief has been greatly helped along by the more extreme elements in the non-West who have often made wild claims."

"In sharp contrast, all Indian systems of philosophy, without any exception, accept the empirical (pratyaksa) as the first means of proof (pramana) while the Lokayata reject inference/deduction as unreliable. So, Indian philosophy considered empirical proof as more reliable than logical inference.Thus, the contrary idea of metaphysical proof as “stronger” than empirical proof would lead at one stroke to the rejection of all Indian systems of philosophy. This illustrates how the metaphysics of formal math is not universal but is biased against other systems of philosophy."

"However, those precise trigonometric values were calculated by Indian mathematicians using infinite series expansions (today called “Taylor's” expansion, “Leibniz” series, etc.), and sophisticated techniques to sum infinite series. These techniques were not comprehended by European mathematicians (who were, then, still struggling at the level of decimal fractions introduced by Stevin, only in 1582). The key difficulty was with the notion of infinite sums, as in the non-terminating, non- recurring decimal expansion for the number pi. The notion of infinity brought religious beliefs prominently into play."

"Another piece of non-textual evidence is the calendar. Because of their arithmetic backwardness, Greeks made a mess of the calendar they had earlier copied from Egypt like their gods. Acknowledging that mess Julius Caesar reformed the Roman calendar with great fanfare, though the net result only aggravated the mess about months (e.g. July has 31 days in honour of Julius, so August competitively has 31 days in honour of Augustus, and February is reduced to 28 or 29)! That (Julian) calendar was adopted as the Christian religious calendar in the 4th c Nicene council to fix the date of the Easter ritual, then the main church festival. However, even that "reformed" calendar had the wrong length of the year (as 365¼ days). That was a gross error even in comparison with 3rd c calendars from India. The gross error arose because the Roman system of numeration had no way to articulate fractions, except for simple fractions like half and quarter; therefore they were unable to state the true length of the year (but that wrong figure is what the colonially educated still learn!). This error (in the second place after the decimal point) naturally led to a noticeable slip in the date of Easter within a century. The church repeatedly tried to correct the error, but even the 5th c Hilarius reforms failed. The church controlled the Roman state then, and Hilarius was a pope, so the only possible reason for this persistent failure to fix the error in the date of the key religious ritual was this : basic knowledge of astronomy was unavailable in the Roman empire. Thus the non-textual evidence states the real hilarious story of Roman incompetence in astronomy, contrary to the tall tale of a Graeco-Roman Ptolemy who authored an advanced text on astronomy in the 2nd c. That is, neither "Claudius Ptolemy" nor advanced knowledge of astronomy existed anywhere in the Roman empire in the 5th c. Lack of accurate knowledge of so basic a parameter as the length of the year nails those false claims?"

"In the Mahabharata, we find the story of Nala and Damayanti. Damayanti announces her intention to remarry by choosing a husband (swayamvara). As Nala and Rituparna (the king of Ayodhya) are rushing from Ayodhya to Vidarbha to participate, they stop near a Vibhitaka tree—the five-faced fruits of which were used in the ancient Indian game of dice. Rituparna shows off his knowledge of statistics by saying: “The number of fruits in the two branches of the tree is 2095, count them if you like.” Nala says he will do exactly that – count them by the empirical method of physically cutting down the tree. Anxious not be delayed, Rituparna dissuades Nala by offering to explain how it was done using sampling and probability theory, also used in the game of dice."

"To summarise, there were different ways to measure angles very accurately in Indian tradition. An angle was defined in the sophisticated way as the length of a curved line, not in the naïve way as something (what thing?) made by two straight lines meeting at a point. The reference to 360 and 720 as a way to measure revolutions is indeed found in the Rgveda, and relates to astronomy and the calendar. Texts like Vedanga Jyotisa (– 1500 CE) use more accurate measures of angles in fractions of degrees. Similar accuracy in angle measurement was part of navigational and astronomical practice."

"Clearly, Macaulay saw education as the most powerful (and cheapest) counter-revolutionary tool. ...Regrettably, few have bothered to study or theorise about Western education as a counter-revolutionary tool."

"This then is the real meaning of those claims of "discovery" by Vasco, Columbus and Cook: people are asked to glorify and celebrate the genocide of non-Christians on three continents. That sets the attitudes of a large mass of people today. Thus, deliberately false historical claims of "discovery" continue to assist the genocidal church politics of world power."

"Adopting this unscientific Christian Gregorian calendar ruins India's economic interests."

"The mathematical theory of probability begins with the theory of permutations and combinations, needed to calculate probabilities in games of chance, such as dice or cards. The earliest account of this theory is found in India. This theory is tied to the theory of the Vedic metre (and the theory of Indian music, in general)."

"We should change the teaching of math, and teach normal math solely for its practical value."

"Could the similarity between the Indian and the “Aristotelian” syllogism be due to transmission? Certainly, there were regular contacts between India and Greece from before the time of Aristotle, as recounted by Herodotus or as evidenced by Alexander’s attempt to find the sea route to India after his army mutinied at the frontiers of India."

"The Indian origin of infinite series, found in widely distributed texts, has long been publicly known to Western scholars (Whish 1832). Recent research (Raju 2007) has shown that these Indian developments really did amount to the calculus. (This brings to the forefront various epistemological issues, and the very philosophy of mathematics taken for granted in Western discourse.) This research has also pushed the historical origin of the calculus in India much further back, to the 5th c. CE Āryabhat.a, and his method of obtaining sine values by numerically solving the corresponding differential equation using a finite difference technique."

"Zeroism is an alternative philosophy of mathematics,1 based on śūnyavāda, a realistic philosophy often ascribed to the Buddhist teacher Nagarjuna (2nd c. CE).2 It is now called zeroism to emphasize that the concern is with the practical and contemporary benefits of that śūnyavāda philosophy, as distinct from fidelity to this or that interpretation of the textual sources of śūnyavāda, which have often been misunderstood and mangled by scholars unfamiliar with the idiom. Indeed, the whole idea of relying on the authority of textual sources is a practice of scriptural traditions, and contrary to śūnyavāda, which denies the validity of proof by authority."

"In Indian philosophy only empirical proof (प्रतरक प्रमार) was universally accepted, by all schools of philosophy. Further, the Lokayata accepted only empirical proofs; specifically they rejected deductive proof (अिम ाि) as inferior. As the Lokayata critique of deductive proofs shows, and as even formal mathematicians today admit, deductively proven theorems are, at best, true relative to postulates. Hence, mere deductive proof does NOT lead to valid knowledge (the goal of Indian philosophy) until the postulates are empirically validated, as in science."

"While there is nothing Vedic in “Vedic mathematics”, there is church dogma in formal mathematics."

"But, the original Indian understanding of calculus as a method of numerically solving differential equations,22 together with non-Archimedean arithmetic and the philosophy of zeroism makes calculus easy enough to teach in five days.24 This ease enables students to solve harder problems such as elliptic integrals required for the first science experiment with the simple pendulum."

"The key thing to recognise here is that false history was (and remains) a key source of colonial power."

"Of course, formal Western mathematics (and indeed much of Western philosophy) is likely to be a long-term casualty of any departure from 2-valued logic. In fact, the very idea that logic (or the basis of probability) is not culturally universal, and may not be empirically certain, unsettles a large segment of Western thought, and its traditional beliefs about induction and deduction."

"The linguistic error of translation in the term “sine” was accompanied by a conceptual error, as in the very word “trigonometry” where the functions relate to the circle, not the triangle. That error persists to this day in the teaching of “trigonometry” which is stuck in the pre-Āryabhat.a era. The word “trigonometry” is in quotes, since this geometric method wrongly suggests that the concepts of sine and cosine relate to the triangle, whereas they actually relate to the circle."

"The translated Indian texts would naturally have gone first to the Jesuit general. There is ample circumstantial evidence that did happen. Christoph Clavius, who authored the Gregorian calendar reform, also published in his name a table of sines in 1607. Curiously, these were the so-called Rsines, in that they explicitly involved the radius of the circle. Simon Stevin follows the same practice for his secant tables. Curiously, Clavius used the same large number for the radius as used in Madhava’s values (Clavius 1607). Documentary evidence of a connection comes from Clavius’ student Matteo Ricci who visited Cochin just prior to the Gregorian reform to get information about Indian methods of timekeeping (Ricci 1581). The Indian timekeeping or astronomy texts near Cochin contained detailed accounts of the calculus. On the epistemic test, those who copy don’t fully understand what they copy. This is also evidence of transmission: Clavius got the imported sine values explicitly interpolated to build a larger table, but did not know enough trigonometry to calculate the size of the earth. Recall that this size was routinely mentioned in Indian texts, and that the size of the earth was a key parameter needed for determining longitudes. (The calendar reform only settled the problem of latitudes.)"

"There is other circumstantial evidence of transmission of calculus to Europe. Clavius’ contemporary, Julius Scaliger, is credited with Julian day-number system which is the same as the Indian ahargan.a. Likewise, another contemporary Tycho Brahe, Royal Astronomer to the Holy Roman Empire, produced the Tychonic astronomical model (in which all planets go round the Sun, which itself goes round the earth) which is just a carbon copy of the astronomical model of Nı̄lakant.ha, stated in his Tantrasangraha. Tycho’s masonry instruments (copied from Ulugh Beg’s Samarkand observatory) were not accurate enough to make accurate observations of Mars, such as made by Parameswaran over a 50 year period. Nevertheless, Tycho, in those days of the Inquisition, kept some secret documents with which his assistant Kepler decamped, after Tycho’s untimely death or murder. Why did Tycho keep mere observations such a secret from his own assistant? How did Kepler, a nearly blind person, arrive at those super-accurate observations, without appropriate instruments?"

"Specifically, time machines are impossible, since realistic time travel implies spontaneity (different from chance). The novel features of this model can be expected to be especially prominent at the microphysical level of biological macromolecules and single particles."

"But how trustworthy is that authority? Are mathematicians more honest than church priests? Uncritical acceptance of authority always invites abuse of authority, and mathematicians are no exception."

"Since there is no evidence for “Pythagoras”, the terminology of the “Pythagorean theorem” is defended by “myth jumping” successively to each of the myths of the person “Euclid” and the myth of the “Euclid” book, that it has axiomatic proofs. But both those myths are false; there is ample counter-evidence against both myths."

"But Hoodbhoy declares the belief in “laws” to be the basis of physics because of his ideological and colonial commitment to slavish imitation of Christian superstitions about laws of nature, an ideology he wants to force on people using the authority of science, just like Macaulay. What he is using is just a modification of the preacher’s doomsday argument (“Covid is round the corner; repent and uncritically accept the authority of science”). Scientists are not more honest than other humans: there are any number of scientists who were and are rascals, just as there are any number of doctors today who are commercialised and dishonest. One uncritically trusts their authority at one’s peril. One can understand why Imran Khan, in a televised debate, got irritated enough to ask Hoodbhoy what he was paid for his propaganda!"

"Let us understand one easy implication of this. The claim that the “Greek” proof of the “Pythagorean theorem” in the “Euclid” book is “superior” to its Indian proof is complete balderdash, though lots of “reliable sources” have asserted it. The actual “Euclid” book uses the same principles of proof as the Indian notion of proof, but is only a lot more prolix."

"It was much later that I realized that most people conflate formal mathematics with the kindergarten mathematics they learned. Thus, they are ignorant of this divorce of formal mathematics from the empirical, though this is stated even at the level of the class IX Indian school text."

"To put matters bluntly, I come from a tradition where, even over 2500 years ago, in supposedly barbaric pre-Christian times, the followers of Buddha and Mahavira were ferociously debating whether it was ethical to unintentionally step on an ant and kill it (hence Jain monks wear masks to avoid unintentionally swallowing any tiny creature, or insect, and carry brooms to sweep aside any ants etc. in their path, to avoid “unintentionally” stepping on them and killing them). So the question is really: what was the ethical or moral justification in the West for mass murder and mass slavery of human beings?"

"However, there is a further problem with deductive proofs. Mathematical theorems, even if validly proved, are invalid knowledge. Hence, the people's philosophers (Lokayata) from India rejected deduction as fallible thousands of years before the church declared it as infallible! The Lokayata objection was simple: deduction may begin from false premises. The classic Lokayata example (SURI 2000) was that observing a wolf’s footprints, people wrongly inferred that a wolf was around, when in actual fact the wolf’s footprints were made at night by a man to demonstrate the fallibility of deductive inference."

"But the primary rule of Western faith-based history of math, as one should well understand by now, is that myth is evidence, and all evidence contrary to the myth, even if this is evidence in front of one's eyes, should be thrown out to preserve the myth."

"This plays on the psychology of the colonized, who are persistently taught that only Western sources are reliable, a cardinal principle of Wikipedia even today."

"Just as an iron ball surrounded by pieces of magnet does not fall though standing (supportless) in the sky, in the same way this (globe of the) Earth though supportless does not fall as it is prevented by (the attraction of) the stars and planets."

"Just as a house lizard runs about on the surface of a pitcher lying in open space, so do the human beings move about comfortably all around the Earth."