Mathematicians From India

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

"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."

"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."

"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."

"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.”"

"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."

"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."

"... 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 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."

"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."

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

"tribhujasya phalashariram samadalakoti bhujardhasamvargah"

"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."

"caturadhikam śatamaṣṭaguṇam dvāṣaṣṭistathā sahasrāṇām ayutadvayaviṣkambhasyāsanno vṛttapariṇāhaḥ."

"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."

"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."

"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."

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

"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."

"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."

"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"

"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."

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

"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."

"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."

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

"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."

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

"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."

"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."

"...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."

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

"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."

"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."

"[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."

"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."

"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)."

"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."

"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."

"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."

"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."

"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."

"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."

"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."

"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."

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

"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."

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

"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."

"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."