Mathematicians From India

"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."

"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."

"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."

"“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."

"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."

"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."

"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 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!"

"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."

"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."

"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."

"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."

"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."

"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."

"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."

"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."

"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."

"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."

"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”."

"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 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 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)."

"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."

"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.’"

"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?"

"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."

"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]."

"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."

"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."

"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]"

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

"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."

"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?"

"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?"

"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."

"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 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 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)."

"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)"