Hindus From India

"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 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."

"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.""

"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."

"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."

"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..."

"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."

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

"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."

"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."

"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."

"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."

"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."

"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..."

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

"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."

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

"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."

"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 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."

"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."

"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."

"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."

"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."

"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... '"

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

"My power is immeasurable; My truth inexplicable, unfathomable."

"The topmost elite of Indian society were also endorsing him. The Indian President, the Prime Minister, the highest judge of the county, governors, chief ministers, members of parliament, vice chancellors of universities, super rich businessmen from Hongkong or America, scientists from all over the world—they all came to see Sai Baba during the seven years I lived in the ashram. He was, as it were, a publicly certified saint. Can so many people go wrong in their judgment regarding his genuineness? I did not think so. I was by now 100 per cent sure that Sai Baba was what he claimed to be—a full avatar—and was glad that I had the good fortune to have him as my guru. To share my guru with millions of others was not what I had dreamt of but, since Sai Baba was extraordinary special, I felt it was great that so many people recognised him as an avatar. We were lucky, because it was not easy, said Sai Baba himself, to entrust oneself to somebody who was still alive. It is much easier, if someone has passed away already."

"In passing away of Sri Sathya Sai Baba the country has lost an iconic spiritual leader and a philanthropist."

"The news of the death of Sai Baba of Puttaparthi has greatly pained me. He was a spiritual person on whom lakhs of people had faith. In his lifetime, he inspired the people of this country and those abroad towards spiritualism and religion."

"Baba has always been my mentor and it was a great loss when he left us. But I remember his lessons and will continue to adhere to them. I felt very low on the day that I sat near his body."

"He's been the moving force behind my life and the life of my family. For us he is God almighty and therefore we are pained."

"I learnt of the passing away of Sri Satya Saibaba with deep and profound sadness. Sri Satya Saibaba was a spiritual leader who inspired millions to lead a moral and meaningful life even as they followed the religion of their choice. His teachings were rooted in the universal ideals of truth, right conduct, peace, love and non-violence... He believed that it is the duty of every person to ensure that all people have access to the basic requirements for sustenance of life. Sri Satya Saibaba was an inspiration to people of all faiths. The passing away of Sri Satya Saibaba is an irreparable loss to all and the nation deeply mourns his passing away..."

"Baba's words and actions have lit the lamp of love in the hearts of devotees and they will always remember him for the good things he has done."

"When such a spiritual personality passes away, we do not pray for their Satgati (divine beatitude) or Atma Shanti (Supreme Peace). On the contrary, we pray to them for our peace and happiness, for they have attained all that and more even when they are in the physical bodies. Mankind will be ever greatful to Babaji for making their lives more beautiful and worth living. May Babaji's blessing continue to shower on one and all alike!"

"It would be simpler to say that the more you limit yourself the more normal you are and the more you expand yourself the more special you are. Here the implication of the word expansion goes beyond one’s own body and immediate family, say, maybe, expanding till your neighbourhood, colony, city, country, world... the entire creation. Expansion is directly proportional to your level of evolution — the more evolved you are, the more expanded you are and the more you will care about your surroundings. Satya Sai Baba was one such evolved, expanded human being... A person who is in vairagya, which is called one of the finalities in Sanatan Kriya, is least concerned about critics or acclaims. He just moves on the chosen path. It’s of no consequence to him who is saying what about him... good or bad. It’s his mission which is important and of utmost concern to him. No matter who says what, no matter what the circumstance, no matter what the occasion, he finds some way to put his point across and that too very effectively."

"Bhagawan Sri Sathya Sai Baba, as he was known, is one of the great spiritual luminaries of the 21st century. Even after his Mahasamadhi, his life and teachings will continue to inspire and influence people to live a noble life. The monumental work that he initiated and achieved is indeed exemplary. In Chinmaya Mission we have many fond memories of our Pujya Gurudev Swami Chinmayananda relating his many meetings with Bhagawan, and the love and regard they shared for each other. We pay our reverential homage to him."

"Sri Sathya Sai Baba will continue to live in the hearts of millions of devotees. His message of 'Love All and Serve All' will resonate in the world forever. His messages have always been universal; especially his message of 'Satya Dharma Shanti Prema' which has transcended all barriers of caste and religion. As Baba has departed earlier than he had predicted, he will return to the planet earlier."

"As a proponent of egalitarian values, he strived to create a world free of the divisions of caste, creed, religion, race and gender. His social initiatives, notably in the fields of education and public health, have been a great service to the world. His demise brings to a close an enigmatic persona that has been a great inspiration."

"The demise of the body was inevitable. But like the sun, which neither rises nor sets, consciousness was indestructible. The message of love and compassion given by Sai Baba had the power to inspire all. Through the implementation of his teachings, he would live forever. May his devotees be able to follow the path of truth and dharma shown by him"

"Elevated souls like Sathya Sai Baba do not come on this earth plane often. Unborn generations to come will wonder at and be amazed by the grandeur of the individual that Baba was and will revere him as a benefactor of humanity. His overpowering love mesmerised and drew to him thousands from all over the world. His giving nature, his caring and sharing made him focus his energies towards ameliorating the suffering of others. For him to live was to love. His enigmatic and charismatic persona will hold sway over the generations to come. His teachings will continue to guide and assist thousands. The world has been left a richer and a better place by having been blessed by the holy presence of Sathya Sai Baba."

"Sathya Sai Baba was an avatar of compassion, love, and selfless service for all mankind. He made no distinction between human beings. Thousands of poor people undergo treatment at his super-specialty hospital in Puttapurthi free of cost. That is the ultimate sewa; for people who cannot afford expensive treatment. Sai Baba's high ideals will show mankind the way for ages to come. His followers will always feel blessed with his emotional and spiritual proximity at all times."

"Leadership is Idealism in Action. The Avatar comes to reveal man to himself, to restore him his birthright of Atmic (Divine) Bliss. He does not come to found a new creed, to breed a new faction, to install a new god. The Avatar comes as Man in order to demonstrate that Man is Divine, in order to be within reach of man. I have come to restore Love among mankind, to cleanse it of narrowness, and restrictive attitudes. That is the main task in the revival of Dharma (Righteousness). Be assured that the Lord has come to save the world from calamity. Your duty is to keep calm, and pray for the happiness and prosperity of all. You cannot be happy when the rest of mankind is unhappy. I offer help; never receive it. My Hand always gives; it never takes. Conclude from this that this must be Divine, not human power. I do not mention about Sai Baba in any of My discourses, though I bear the name as Avatar of Sai Baba. I do not appreciate in the least the distinction between the varied appearances of God: Sai, Rama, Krishna, Christ, Nanak, Buddha etc. I do not proclaim that this is more important or that the other is less important. Continue your worship of your chosen God along the lines already familiar to you. Then you will find that you are coming nearer and nearer to Me, for all names are Mine and all forms are Mine. There is no need to change your chosen God and adopt a new one when you have seen Me and heard Me. Come just one step forward, I shall take a hundred steps towards you. Shed just one tear, I shall wipe a hundred from your eyes. I bless only thus: May your Ananda, may your Bliss grow. I have come to give you the Key of the treasure of Ananda or Bliss, to tell you how to tap that spring, for you have forgotten the way to blessedness. I never exult when I am extolled, nor shrink when I am reviled. Few have realized My Purpose and Significance, but I am not worried. When things that are not in Me are attributed to Me, why should I worry? When things that are in Me are mentioned, why should I exult? For Me, it is always, "Yes, yes, yes." IF you give up all and surrender to the Lord, He will guard you and guide you. The Lord has come just for this very task. Many hesitate to believe that things will improve, that life will be happy for all and full of joy, and that the Golden Age will recur. Let me assure you that this Dharmaswarupa, this divine body, has not come in vain. It will succeed in averting the crisis that has come upon humanity."

"Prema (love): practise that; develop that; spread that; and all the hatreds and jealousies of today will disappear. That is the duty of the Divine Life Society, here as well as elsewhere."

"Be sincere; talk only about your genuine experience; do not distort, exaggerate or falsify that experience."

"As a matter of fact, there is no trace of ill-health in Me. I am always healthy. Not only today, till 96 years I will be like this."

"As the attraction power is increasing in Me... you are going to witness many astounding events in future... Even while I am speaking to you now, My feet are being pulled by the earth. If I lift one foot, the earth attracts the other foot. Whatever I touch with My hand, it also gets attracted. This magnetic power is present in every man... I try to lift My foot but it is very difficult."

"Some people think, Swami calls himself a powerful magnet, but He is unable to attract us. The fault lies in them only."