Mathematicians From India

"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."

"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."

"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?"

"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."

"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)."

"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)."

"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."

"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)."

"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."

"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 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)."

"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."

"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?"

"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."

"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."

"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."

"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 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)"

"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."

"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."

"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."

"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)."

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

"“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.”"

""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)."

"“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”."

"“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.”"

"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.”"

"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!)"

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

"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."

"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."

"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."

"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)."

"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."

"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)."

"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”."

"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."