In this chapter we give the verbatim opinion of mathematicians and experts about Vedic Mathematics in their articles, that have appeared in the print media. The article of Prof. S.G. Dani, School of Mathematics, Tata Institute of Fundamental Research happen to give a complete analysis of Vedic Mathematics.

We have given his second article verbatim because we do not want any bias or our opinion to play any role in our analysis

[32].

However we do not promise to discuss all the articles. Only articles which show “How Vedic is Vedic Mathematics?” is given for the perusal of the reader. We thank them for their articles and quote them verbatim. The book on Vedic Mathematics by Jagadguru Sankaracharya of Puri has been translated into Tamil by Dr. V.S. Narasimhan, a Retired Professor of an arts college and C. Mailvanan, M.Sc Mathematics (Vidya Barathi state-level Vedic Mathematics expert) in two volumes. The first edition appeared in 1998 and the corrected second edition in 2003.

In Volume I of the Tamil book the introduction is as follows: “Why was the name Vedic Mathematics given? On the title “a trick in the name of Vedic Mathematics” though professors in mathematics praise the sutras, they argue that the title Vedic Mathematics is not well suited. According to them

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the sutras published by the Swamiji are not found anywhere in the Vedas. Further the branches of mathematics like algebra and calculus which he mentions, did not exist in the Vedic times. It may help school students but only in certain problems where shortcut methods can be used. The Exaggeration that, it can be used in all branches of mathematics cannot be accepted.

Because it gives answers very fast it can be called “speed maths”. He has welcomed suggestions and opinions of one and all.

It has also become pertinent to mention here that Jagadguru Puri Sankaracharya for the first time visited the west in 1958. He had been to America at the invitation of the Self Realization Fellowship Los Angeles, to spread the message of Vedanta. The book Vedic Metaphysics is a compilation of some of his discourses delivered there. On 19 February 1958, he has given a talk and demonstration to a small group of student mathematicians at the California Institute of Technology, Pasadena, California.

This talk finds its place in chapter XII of the book Vedic Metaphysics pp. 156-196 [52] most of which has appeared later on, in his book on Vedic Mathematics [51]. However some experts were of the opinion, that if Swamiji would have remained as Swamiji ‘or’ as a ‘mathematician’ it would have been better. His intermingling and trying to look like both has only brought him less recognition in both Mathematics and on Vedanta. The views of Wing Commander Vishva Mohan Tiwari, under the titles conventional to unconventionally original speaks of Vedic Mathematics as follows:

“Vedic Mathematics mainly deals with various Vedic mathematical formulas and their applications of carrying out tedious and cumbersome arithmetical operations, and to a very large extent executing them mentally. He feels that in this field of mental arithmetical operations the works of the famous mathematicians Trachtenberg and Lester Meyers (High speed mathematics) are elementary compared to that of Jagadguruji … An attempt has been made in this note to explain the unconventional aspects of the methods. He then gives a very brief sketch of first four chapters of Vedic Mathematics”.

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This chapter has seven sections; Section one gives the verbatim analysis of Vedic Mathematics given by Prof. Dani in his article in Frontline [31].

A list of eminent signatories asking people to stop this fraud on our children is given verbatim in section two. Some views given about the book both in favour of and against is given in section three.

Section four gives the essay Vedas: Repositories of ancient lore. “A rational approach to study ancient literature” an article found in Current Science, volume 87, August 2004 is given in Section five. Section Six gives the “Shanghai Rankings and Indian Universities.” The final section gives conclusion derived on Vedic Mathematics and calculation of Guru Tirthaji.

**2.1**** Views of Prof. S.G. Dani about Vedic Mathematics ****from**** Frontline**

Views of Prof. S.G.Dani gave the authors a greater technical insight into Vedic Mathematics because he has written 2 articles in Frontline in 1993. He has analyzed the book extremely well and we deeply acknowledge the services of professor S.G.Dani to the educated community in general and school students in particular. This section contains the verbatim views of Prof. Dani that appeared in Frontline magazine. He has given a marvelous analysis of the book Vedic Mathematics and has daringly concluded.

“One would hardly have imagine that a book which is transparently not from any ancient source or of any great mathematical significance would one day be passed off as a storehouse of some ancient mathematical treasure. It is high time saner elements joined hands to educate people on the truth of this so-called Vedic Mathematics and prevent the use of public money and energy on its propagation, beyond the limited extent that may be deserved, lest the intellectual and educational life in the country should get vitiated further and result in wrong attitudes to both history and mathematics, especially in the coming generation.”

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**Myths and Reality: On ‘Vedic Mathematics’**

S.G. Dani, School of Mathematics, Tata Institute of Fundamental Research

*An updated version of the 2-part article in Frontline, 22 Oct. and 5 Nov. 1993*

We in India have good reasons to be proud of a rich heritage in science, philosophy and culture in general, coming to us down the ages. In mathematics, which is my own area of specialization, the ancient Indians not only took great strides long before the Greek advent, which is a standard reference point in the Western historical perspective, but also enriched it for a long period making in particular some very fundamental contributions such as the place-value system for writing numbers as we have today, introduction of zero and so on. Further, the sustained development of mathematics in India in the post-Greek period was indirectly instrumental in the revival in Europe after “its dark ages”.

Notwithstanding the enviable background, lack of adequate attention to academic pursuits over a prolonged period, occasioned by several factors, together with about two centuries of Macaulayan educational system, has unfortunately resulted, on the one hand, in a lack of awareness of our historical role in actual terms and, on the other, an empty sense of pride which is more of an emotional reaction to the colonial domination rather than an intellectual challenge. Together they provide a convenient ground for extremist and misguided elements in society to “reconstruct history” from nonexistent or concocted source material to whip up popular euphoria.

That this anti-intellectual endeavour is counter-productive in the long run and, more important, harmful to our image as a mature society, is either not recognized or ignored in favour of short-term considerations. Along with the obvious need to accelerate the process of creating an awareness of our past achievements, on the strength of authentic information, a more urgent need has also arisen to confront and expose such baseless constructs before it is too late. This is not merely a question of setting the record straight. The motivated versions have a way of corrupting the intellectual processes in society and weakening their very foundations in the long run, which needs to be prevented at all costs. The so-called “Vedic Mathematics”

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is a case in point. A book by that name written by Jagadguru Swami Shri Bharati Krishna Tirthaji Maharaja (Tirthaji, 1965) is at the centre of this pursuit, which has now acquired wide following; Tirthaji was the Shankaracharya of Govardhan Math, Puri, from 1925 until he passed away in 1960. The book was published posthumously, but he had been carrying out a campaign on the theme for a long time, apparently for several decades, by means of lectures, blackboard demonstrations, classes and so on. It has been known from the beginning that there is no evidence of the contents of the book being of Vedic origin; the Foreword to the book by the General Editor, Dr. A.S.Agrawala, and an account of the genesis of the work written by Manjula Trivedi, a disciple of the swamiji, make this clear even before one gets to the text of the book. No one has come up with any positive evidence subsequently either.

There has, however, been a persistent propaganda that the material is from the Vedas. In the face of a false sense of national pride associated with it and the neglect, on the part of the knowledgeable, in countering the propaganda, even educated and well meaning people have tended to accept it uncritically. The vested interests have also involved politicians in the propaganda process to gain state support. Several leaders have lent support to the “Vedic Mathematics” over the years, evidently in the belief of its being from ancient scriptures. In the current environment, when a label as ancient seems to carry considerable premium irrespective of its authenticity or merit, the purveyors would have it going easy.

Large sums have been spent both by the Government and several private agencies to support this “Vedic Mathematics”, while authentic Vedic studies continue to be neglected. People, especially children, are encouraged to learn and spread the contents of the book, largely on the baseless premise of their being from the Vedas. With missionary zeal several “devotees” of this cause have striven to take the “message” around the world; not surprisingly, they have even met with some success in the West, not unlike some of the gurus and yogis peddling their own versions of “Indian philosophy”. Several people are also engaged in “research” in the new “Vedic Mathematics.”

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To top it all, when in the early nineties the Uttar Pradesh Government introduced “Vedic Mathematics” in school text books, the contents of the swamiji’s book were treated as if they were genuinely from the Vedas; this also naturally seems to have led them to include a list of the swamiji’s sutras on one of the opening pages (presumably for the students to learn them by heart and recite!) and to accord the swamiji a place of honour in the “brief history of Indian mathematics” described in the beginning of the textbook, together with a chart, which cu- riously has Srinivasa Ramanujan’s as the only other name from the twentieth century!

For all their concern to inculcate a sense of national pride in children, those responsible for this have not cared for the simple fact that modern India has also produced several notable mathematicians and built a worthwhile edifice in mathematics

(as also in many other areas). Harish Chandra’s work is held in great esteem all over the world and several leading seats of learning of our times pride themselves in having members pursuing his ideas; (see, for instance, Langlands, 1993). Even among those based in India, several like Syamdas Mukhopadhyay, Ganesh Prasad, B.N.Prasad, K.Anand Rau, T.Vijayaraghavan, S.S.Pillai, S.Minakshisundaram, Hansraj Gupta, K.G.Ramanathan, B.S.Madhava Rao, V.V.Narlikar, P.L.Bhatnagar and so on and also many living Indian mathematicians have carved a niche for themselves on the international mathematical scene (see Narasimhan, 1991). Ignoring all this while introducing the swamiji’s name in the

“brief history” would inevitably create a warped perspective in children’s minds, favouring gimmickry rather than professional work. What does the swamiji’s “Vedic Mathematics” seek to do and what does it achieve? In his preface of the book, grandly titled” A Descriptive Prefatory Note on the astounding Wonders of Ancient Indian Vedic Mathematics,” the swamiji tells us that he strove from his childhood to study the Vedas critically “to prove to ourselves (and to others) the correctness (or otherwise)”of the “derivational meaning” of “Veda” that the” Vedas should contain within themselves all the knowledge needed by the mankind relating not only to spiritual matters but also those usually described as purely ‘secular’, ‘temporal’ or

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‘worldly’; in other words, simply because of the meaning of the word ‘Veda’, everything that is worth knowing is expected to be contained in the vedas and the swamiji seeks to prove it to be the case!

It may be worthwhile to point out here that there would be room for starting such an enterprise with the word ‘science’! He also describes how the “contemptuous or at best patronising ” attitude of Orientalists, Indologists and so on strengthened his determination to unravel the too-long-hidden mysteries of philosophy and science contained in ancient India’s Vedic lore, with the consequence that, “after eight years of concentrated contemplation in forest solitude, we were at long last able to recover the long lost keys which alone could unlock the portals thereof.”

The mindset revealed in this can hardly be said to be suitable in scientific and objective inquiry or pursuit of knowledge, but perhaps one should not grudge it in someone from a totally different milieu, if the outcome is positive. One would have thought that with all the commitment and grit the author would have come up with at least a few new things which can be attributed to the Vedas, with solid evidence. This would have made a worthwhile contribution to our understanding of our heritage. Instead, all said and done there is only the author’s certificate that “we were agreeably astonished and intensely gratified to find that exceedingly though mathematical problems can be easily and readily solved with the help of these ultra-easy Vedic sutras (or mathematical aphorisms) contained in the Parishishta (the appendix portion) of the Atharva Veda in a few simple steps and by methods which can be conscientiously described as mere ‘mental arithmetic’ ”(paragraph 9 in the preface). That passing reference to the Atharva Veda is all that is ever said by way of source material for the contents. The sutras, incidentally, which appeared later scattered in the book, are short phrases of just about two to four words in Sanskrit, such as Ekadhikena Purvena or Anurupye Shunyam Anyat. (There are 16 of them and in addition there are 13 of what are called sub-sutras, similar in nature to the sutras).

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The first key question, which would occur to anyone, is where are these sutras to be found in the Atharva Veda. One does not mean this as a rhetorical question. Considering that at the outset the author seemed set to send all doubting Thomases packing, the least one would expect is that he would point out where the sutras are, say in which part, stanza, page and so on, especially since it is not a small article that is being referred to. Not only has the author not cared to do so, but when Prof.K.S.Shukla, a renowned scholar of ancient Indian mathematics, met him in 1950, when the swamiji visited Lucknow to give a blackboard demonstration of his “Vedic Mathematics”, and requested him to point out the sutras in question in the Parishishta of the Atharva Veda, of which he even carried a copy (the standard version edited by G.M.Bolling and J.Von Negelein), the swamiji is said to have told him that the 16 sutra demonstrated by him were not in those Parishishtas and that “they occurred in his own Parishishta and not any other” (Shukla, 1980, or Shukla, 1991). What justification the swamiji thought he had for introducing an appendix in the Atharva Veda, the contents of which are nevertheless to be viewed as from the Veda, is anybody’s guess. In any case, even such a Parishishta, written by the swamiji, does not exist in the form of a Sanskrit text.

Let us suppose for a moment that the author indeed found the sutras in some manuscript of the Atharva Veda, which he came across. Would he not then have preserved the manuscript? Would he not have shown at least to some people where the sutras are in the manuscript? Would he not have revealed to some cherished students how to look for sutras with such profound mathematical implications as he attributes to the sutras in question, in that or other manuscripts that may be found? While there is a specific mention in the write-up of Manjula Trivedi, in the beginning of the book, about some 16volume manuscript written by the swamiji having been lost in 1956, there is no mention whatever (let alone any lamentation that would be due in such an event) either in her write-up nor in the swamiji’s preface about any original manuscript having been lost. No one certainly has come forward with any information received from the swamiji with regard to the other questions

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above. It is to be noted that want of time could not be a factor in any of this, since the swamiji kindly informs us in the preface that “Ever since (i.e. since several decades ago), we have been carrying on an incessant and strenuous campaign for the India- wide diffusion of all this scientific knowledge”.

The only natural explanation is that there was no such manuscript. It has in fact been mentioned by Agrawala in his general editor’s foreword to the book, and also by Manjula Trivedi in the short account of the genesis of the work, included in the book together with a biographical sketch of the swamiji, that the sutras do not appear in hitherto known Parishishtas. The general editor also notes that the style of language of the sutras

“point to their discovery by Shri Swamiji himself ” (emphasis added); the language style being contemporary can be confirmed independently from other Sanskrit scholars as well. The question why then the contents should be considered

‘Vedic’ apparently did not bother the general editor, as he agreed with the author that “by definition” the Vedas should contain all knowledge (never mind whether found in the 20th century, or perhaps even later)! Manjula Trivedi, the disciple has of course no problem with the sutras not being found in the Vedas as she in fact says that they were actually reconstructed by her beloved “Gurudeva,” on the basis of intuitive revelation from material scattered here and there in the Atharva Veda, after

“assiduous research” and ‘Tapas’ for about eight years in the forests surrounding Shringeri.” Isn’t that adequate to consider them to be “Vedic”? Well, one can hardly argue with the devout! There is a little problem as to why the Gurudeva him- self did not say so (that the sutras were reconstructed) rather than referring to them as sutras contained in the Parishishta of the Atharva Veda, but we will have to let it pass. Anyway the fact remains that she was aware that they could not actually be located in what we lesser mortals consider to be the Atharva Veda. The question of the source of the sutras is merely the first that would come to mind, and already on that there is such a muddle. Actually, even if the sutras were to be found, say in the Atharva Veda or some other ancient text, that still leaves open another fundamental question as to whether they mean or yield, in some cognisable way, what the author claims; in other words,

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we would still need to know whether such a source really contains the mathematics the swamiji deals with or merely the phrases, may be in some quite different context. It is interesting to consider the swamiji’s sutras in this light. One of them, for instance, is Ekadhikena Purvena which literally just means “by one more than the previous one.” In chapter I, the swamiji tells us that it is a sutra for finding the digits in the decimal expansion of numbers such as 1/19, and 1/29, where the denominator is a number with 9 in the unit’s place; he goes on to give a page-long description of the procedure to be followed, whose only connection with the sutra is that it involves, in particular, repeatedly multiplying by one more than the previous one, namely 2, 3 and so on, respectively, the “previous one” being the number before the unit’s place; the full procedure involves a lot more by way of arranging the digits which can in no way be read off from the phrase.

In Chapter II, we are told that the same sutra also means that to find the square of a number like 25 and 35, (with five in unit’s place) multiply the number of tens by one more than itself and write 25 ahead of that; like 625, 1,225 and so on. The phrase Ekanyunena Purvena which means “by one less than the previous one” is however given to mean something which has neither to do with decimal expansions nor with squaring of numbers but concerns multiplying together two numbers, one of which has 9 in all places (like 99,999, so on.)!

Allowing oneself such unlimited freedom of interpretation, one can also interpret the same three-word phrase to mean also many other things not only in mathematics but also in many other subjects such as physics, chemistry, biology, economics, sociology and politics. Consider, for instance, the following

“meaning”: the family size may be allowed to grow, at most, by one more than the previous one. In this we have the family- planning message of the 1960s; the “previous one” being the couple, the prescription is that they should have no more than three children. Thus the lal trikon (red triangle) formula may be seen to be “from the Atharva Veda,” thanks to the swamiji’s novel technique (with just a bit of credit to yours faithfully). If you think the three children norm now outdated, there is no need to despair. One can get the two-children or even the one-

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child formula also from the same sutra; count only the man as the “previous one” (the woman is an outsider joining in marriage, isn’t she) and in the growth of the family either count only the children or include also the wife, depending on what suits the desired formula!

Another sutra is Yavadunam, which means “as much less;” a lifetime may not suffice to write down all the things such a phrase could “mean,” in the spirit as above. There is even a sub- sutra, Vilokanam (observation) and that is supposed to mean various mathematical steps involving observation! In the same vein one can actually suggest a single sutra adequate not only for all of mathematics but many many subjects: Chintanam

(think)!

It may be argued that there are, after all, ciphers which convey more information than meets the eye. But the meaning in those cases is either arrived at from the knowledge of the deciphering code or deduced in one or other way using various kinds of contexual information. Neither applies in the present case. The sutras in the swamiji’s book are in reality mere names for various steps to be followed in various contexts; the steps themselves had to be known independently. In other words, the mathematical step is not arrived at by understanding or interpreting what are given as sutras; rather, sutras somewhat suggestive of the meaning of the steps are attached to them like names. It is like associating the ‘sutra’ VIBGYOR to the sequence of colours in rainbow (which make up the white light). Usage of words in Sanskrit, a language which the popular mind unquestioningly associates with the distant past(!), lend the contents a bit of antique finish!

An analysis of the mathematical contents of Tirthaji’s book also shows that they cannot be from the Vedas. Though unfortunately there is considerable ignorance about the subject, mathematics from the Vedas is far from being an unexplored area. Painstaking efforts have been made for well over a century to study the original ancient texts from the point of view of understanding the extent of mathematical knowledge in ancient times. For instance, from the study of Vedic Samhitas and Brahamanas it has been noted that they had the system of counting progressing in multiples of 10 as we have today and

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that they considered remarkably large numbers, even up to 14 digits, unlike other civilizations of those times. From the Vedanga period there is in fact available a significant body of mathematical literature in the form of Shulvasutras, from the period between 800 bc and 500 bc, or perhaps even earlier, some of which contain expositions of various mathematical principles involved in construction of sacrificial ‘vedi’s needed in performing’ yajna’s (see, for instance, Sen and Bag 1983). Baudhyana Shulvasutra, the earliest of the extant Shulvasutras, already contains, for instance, what is currently known as Pythagoras’ Theorem (Sen and Bag, 1983, page 78,

1.12). It is the earliest known explicit statement of the theorem in the general form (anywhere in the world) and precedes Pythagoras by at least a few hundred years. The texts also show a remarkable familiarity with many other facts from the so- called Euclidean Geometry and it is clear that considerable use was made of these, long before the Greeks formulated them. The work of George Thibaut in the last century and that of A.Burk around the turn of the century brought to the attention of the world the significance of the mathematics of the Shulvasutras. It has been followed up in this century by both foreign and Indian historians of mathematics. It is this kind of authentic work, and not some mumbo-jumbo that would highlight our rich heritage. I would strongly recommend to the reader to peruse the monograph, The Sulbasutras by S.N.Sen and A.K.Bag (Sen and Bag, 1983), containing the original sutras, their translation and a detailed commentary, which includes a survey of a number of earlier works on the subject. There are also several books on ancient Indian mathematics from the Vedic period.

The contents of the swamiji’s book have practically nothing in common with what is known of the mathematics from the Vedic period or even with the subsequent rich tradition of mathematics in India until the advent of the modern era; incidentally, the descriptions of mathematical principles or procedures in ancient mathematical texts are quite explicit and not in terms of cryptic sutras. The very first chapter of the book

(as also chapters XXVI to XXVIII) involves the notion of decimal fractions in an essential way. If the contents are to be

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Vedic, there would have had to be a good deal of familiarity with decimal fractions, even involving several digits, at that time. It turns out that while the Shulvasutras make extensive use of fractions in the usual form, nowhere is there any indication of fractions in decimal form. It is inconceivable that such an important notion would be left out, had it been known, from what are really like users manuals of those times, produced at different times over a prolonged period. Not only the Shulvasutras and the earlier Vedic works, but even the works of mathematicians such as Aryabhata, Brahmagupta and Bhaskara, are not found to contain any decimal fractions. Is it possible that none of them had access to some Vedic source that the swamiji could lay his hands on (and still not describe it specifically)? How far do we have to stretch our credulity?

The fact is that the use of decimal fractions started only in the 16th century, propagated to a large extent by Francois Viete; the use of the decimal point (separating the integer and the fractional parts) itself, as a notation for the decimal representation, began only towards the end of the century and acquired popularity in the 17th century following their use in John Napier’s logarithm tables (see, for instance, Boyer, 1968, page 334).

Similarly, in chapter XXII the swamiji claims to give

“sutras relevant to successive differentiation, covering the theorems of Leibnitz, Maclaurin, Taylor, etc. and a lot of other material which is yet to be studied and decided on by the great mathematicians of the present-day Western world;” it should perhaps be mentioned before we proceed that the chapter does not really deal with anything of the sort that would even remotely justify such a grandiloquent announcement, but rather deals with differentiation as an operation on polynomials, which is a very special case reducing it all to elementary algebra devoid of the very soul of calculus, as taught even at the college level.

Given the context, we shall leave Leibnitz and company alone, but consider the notions of derivative and successive differentiation. Did the notions exist in the Vedic times? While certain elements preliminary to calculus have been found in the works of Bhaskara II from the 12th century and later Indian

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mathematicians in the pre-calculus era in international mathematics, such crystallised notions as the derivative or the integral were not known. Though a case may be made that the developments here would have led to the discovery of calculus in India, no historians of Indian mathematics would dream of proposing that they actually had such a notion as the derivative, let alone successive differentiation; the question here is not about performing the operation on polynomials, but of the con- cept. A similar comment applies with regard to integration, in chapter XXIV. It should also be borne in mind that if calculus were to be known in India in the early times, it would have been acquired by foreigners as well, long before it actually came to be discovered, as there was enough interaction between India and the outside world.

If this is not enough, in Chapter XXXIX we learn that analytic conics has an “important and predominating place for itself in the Vedic system of mathematics,” and in Chapter XL we find a whole list of subjects such as dynamics, statics, hydrostatics, pneumatics and applied mathematics listed alongside such elementary things as subtractions, ratios, proportions and such money matters as interest and annuities

(!), discounts (!) to which we are assured, without going into details, that the Vedic sutras can be applied. Need we comment any further on this? The remaining chapters are mostly elementary in content, on account of which one does not see such marked incongruities in their respect. It has, however, been pointed out by Shukla that many of the topics considered in the book are alien to the pursuits of ancient Indian mathematicians, not only form the Vedic period but until much later (Shukla,

1989 or Shukla, 1991). These include many such topics as factorisation of algebraic expressions, HCF (highest common factor) of algebraic expressions and various types of simultaneous equations. The contents of the book are akin to much later mathematics, mostly of the kind that appeared in school books of our times or those of the swamiji’s youth, and it is unthinkable, in the absence of any pressing evidence, that they go back to the Vedic lore. The book really consists of a compilation of tricks in elementary arithmetic and algebra, to be applied in computations with numbers and polynomials. By a

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“trick” I do not mean a sleight of hand or something like that; in a general sense a trick is a method or procedure which involves observing and exploring some special features of a situation, which generally tend to be overlooked; for example, the trick described for finding the square of numbers like 15 and 25 with

5 in the unit’s place makes crucial use of the fact of 5 being half of 10, the latter being the base in which the numbers are written. Some of the tricks given in the book are quite interesting and admittedly yield quicker solutions than by standard methods

(though the comparison made in the book are facetious and misleading). They are of the kind that an intelligent hobbyist ex- perimenting with numbers might be expected to come up with. The tricks are, however, based on well-understood mathematical principles and there is no mystery about them.

Of course to produce such a body of tricks, even using the well-known is still a non-trivial task and there is a serious question of how this came to be accomplished. It is sometimes suggested that Tirthaji himself might have invented the tricks. The fact that he had a M.A.degree in mathematics is notable in this context. It is also possible that he might have learnt some of the tricks from some elders during an early period in his life and developed on them during those “eight years of concentrated contemplation in forest solitude:” this would mean that they do involve a certain element of tradition, though not to the absurd extent that is claimed. These can, however, be viewed only as possibilities and it would not be easy to settle these details. But it is quite clear that the choice is only between alternatives involving only the recent times.

It may be recalled here that there have also been other instances of exposition and propagation of such faster methods of computation applicable in various special situations (without claims of their coming from ancient sources). Trachtenberg’s Speed System (see Arther and McShane, 1965) and Lester Meyers’ book, High-Speed Mathematics (Meyers, 1947) are some well-known examples of this. Trachtenberg had even set up an Institute in Germany to provide training in high-speed mathematics. While the swamiji’s methods are independent of these, for the most part they are similar in spirit.

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One may wonder why such methods are not commonly adopted for practical purposes. One main point is that they turn out to be quicker only for certain special classes of examples. For a general example the amount of effort involved (for instance, the count of the individual operations needed to be performed with digits, in arriving at the final answer) is about the same as required by the standard methods; in the swamiji’s book, this is often concealed by not writing some of the steps involved, viewing it as “mental arithmetic.” Using such methods of fast arithmetic involves the ability or practice to recognize various patterns which would simplify the calculations. Without that, one would actually spend more time, in first trying to recognize patterns and then working by rote anyway, since in most cases it is not easy to find useful patterns. People who in the course of their work have to do computations as they arise, rather than choose the figures suitably as in the demonstrations, would hardly find it convenient to carry them out by employing umpteen different ways depending on the particular case, as the methods of fast arithmetic involve. It is more convenient to follow the standard method, in which one has only to follow a set procedure to find the answer, even though in some cases this might take more time. Besides, equipment such as calculators and computers have made it unnecessary to tax one’s mind with arithmetical computations. Incidentally, the suggestion that this “Vedic Mathematics” of the Shankaracharya could lead to improvement in computers is totally fallacious, since the underlying mathematical principles involved in it were by no means

unfamiliar in professional circles.

One of the factors causing people not to pay due attention to the obvious questions about “Vedic Mathematics” seems to be that they are overwhelmed by a sense of wonderment by the tricks. The swamiji tells us in the preface how “the educationists, the cream of the English educated section of the people including highest officials (e.g. the high court judges, the ministers etc.) and the general public as such were all highly impressed; nay thrilled, wonder-struck and flabbergasted!” at his demonstrations of the “Vedic Mathematics.” Sometimes one comes across reports about similar thrilling demonstrations by

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some of the present-day expositors of the subject. Though inevitably they have to be taken with a pinch of salt, I do not entirely doubt the truth of such reports. Since most people have had a difficult time with their arithmetic at school and even those who might have been fairly good would have lost touch, the very fact of someone doing some computations rather fast can make an impressive sight. This effect may be enhanced with well-chosen examples, where some quicker methods are applicable.

Even in the case of general examples where the method employed is not really more efficient than the standard one, the computations might appear to be fast, since the demonstrator would have a lot more practice than the people in the audience. An objective assessment of the methods from the point of view of overall use can only be made by comparing how many individual calculations are involved in working out various general examples, on an average, and in this respect the methods of fast arithmetic do not show any marked advantage which would offset the inconvenience indicated earlier. In any case, it would be irrational to let the element of surprise interfere in judging the issue of origin of “Vedic Mathematics” or create a dreamy and false picture of its providing solutions to all kinds of problems.

It should also be borne in mind that the book really deals only with some middle and high school level mathematics; this is true despite what appear to be chapters dealing with some notions in calculus and coordinate geometry and the mention of a few, little more advanced topics, in the book. The swamiji’s claim that “there is no part of mathematics, pure or applied, which is beyond their jurisdiction” is ludicrous. Mathematics actually means a lot more than arithmetic of numbers and algebra of polynomials; in fact multiplying big numbers together, which a lot of people take for mathematics, is hardly something a mathematician of today needs to engage himself in. The mathematics of today concerns a great variety of objects beyond the high school level, involving various kinds of ab- stract objects generalising numbers, shapes, geometries, measures and so on and several combinations of such structures, various kinds of operations, often involving infinitely many en-

47

tities; this is not the case only about the frontiers of mathematics but a whole lot of it, including many topics applied in physics, engineering, medicine, finance and various other subjects.

Despite all its pretentious verbiage page after page, the swamiji’s book offers nothing worthwhile in advanced mathematics whether concretely or by way of insight. Modern mathematics with its multitude of disciplines (group theory, topology, algebraic geometry, harmonic analysis, ergodic theory, combinatorial mathematics-to name just a few) would be a long way from the level of the swamiji’s book. There are occasionally reports of some “researchers” applying the swamiji’s “Vedic Mathematics” to advanced problems such as Kepler’s problem, but such work involves nothing more than tinkering superficially with the topic, in the manner of the swamiji’s treatment of calculus, and offers nothing of interest to professionals in the area.

Even at the school level “Vedic Mathematics” deals only with a small part and, more importantly, there too it concerns itself with only one particular aspect, that of faster computation. One of the main aims of mathematics education even at the elementary level consists of developing familiarity with a variety of concepts and their significance. Not only does the approach of “Vedic Mathematics” not contribute anything towards this crucial objective, but in fact might work to its detriment, because of the undue emphasis laid on faster computation. The swamiji’s assertion “8 months (or 12 months) at an average rate of 2 or 3 hours per day should suffice for completing the whole course of mathematical studies on these Vedic lines instead of 15 or 20 years required according to the existing systems of the Indian and also foreign universities,” is patently absurd and hopefully nobody takes it seriously, even among the activists in the area. It would work as a cruel joke if some people choose to make such a substitution in respect of their children.

It is often claimed that “Vedic Mathematics” is well- appreciated in other countries, and even taught in some schools in UK etc.. In the normal course one would not have the means to examine such claims, especially since few details are generally supplied while making the claims. Thanks to certain

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special circumstances I came to know a few things about the St. James Independent School, London which I had seen quoted in this context. The School is run by the ‘School of Economic Science’ which is, according to a letter to me from Mr. James Glover, the Head of Mathematics at the School, “engaged in the practical study of Advaita philosophy”. The people who run it have had substantial involvement with religious groups in India over a long period. Thus in essence their adopting “Vedic Mathematics” is much like a school in India run by a religious group adopting it; that school being in London is beside the point. (It may be noted here that while privately run schools in India have limited freedom in choosing their curricula, it is not the case in England). It would be interesting to look into the background and motivation of other institutions about which similar claims are made. At any rate, adoption by institutions abroad is another propaganda feature, like being from ancient source, and should not sway us.

It is not the contention here that the contents of the book are not of any value. Indeed, some of the observations could be used in teaching in schools. They are entertaining and could to some extent enable children to enjoy mathematics. It would, however, be more appropriate to use them as aids in teaching the related concepts, rather than like a series of tricks of magic. Ultimately, it is the understanding that is more important than the transient excitement, By and large, however, such pedagogical application has limited scope and needs to be made with adequate caution, without being carried away by motivated propaganda.

It is shocking to see the extent to which vested interests and persons driven by guided notions are able to exploit the urge for cultural self-assertion felt by the Indian psyche. One would hardly have imagined that a book which is transparently not from any ancient source or of any great mathematical significance would one day be passed off as a storehouse of some ancient mathematical treasure. It is high time saner elements joined hands to educate people on the truth of this so- called Vedic Mathematics and prevent the use of public money and energy on its propagation, beyond the limited extent that may be deserved, lest the intellectual and educational life in the

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country should get vitiated further and result in wrong attitudes to both history and mathematics, especially in the coming generation.

**References**

[1] Ann Arther and Rudolph McShane, *The Trachtenberg Speed System of Basic Mathematics (English edition)*, Asia Publishing House, New Delhi, 1965.

[2] Carl B. Boyer, *A History of Mathematics*, John Wiley and

Sons, 1968.

[3] R.P. Langlands, *Harish-Chandra (11 October 1923 -16*

*October 1983)*, Current Science, Vol. 65: No. 12, 1993.

[4] Lester Meyers, *High-Speed Mathematics*, Van Nostrand, New York, 1947.

[5] Raghavan Narasimhan, *The Coming of Age of Mathematics in India*, Miscellanea Mathematica, 235–258, Springer- Verlag, 1991.

[6] S.N. Sen and A.K. Bag, *The Sulbasutras*, Indian National

Science Academy, New Delhi, 1983. .

[7] K.S. Shukla, *Vedic Mathematics — the illusive title of Swamiji’s book*, Mathematical Education, Vol 5: No. 3, January-March 1989.

[8] K.S. Shukla, *Mathematics — The Deceptive Title of Swamiji’s Book*, in Issues in Vedic Mathematics, (ed: H.C.Khare), Rashtriya Veda Vidya Prakashan and Motilal Banarasidass Publ., 1991.

[9] Shri Bharati Krishna Tirthaji, *Vedic Mathematics*, Motilal

Banarasidass, New Delhi, 1965.

**2.2**** Neither Vedic Nor Mathematics**

We, the undersigned, are deeply concerned by the continuing attempts to thrust the so-called `Vedic Mathematics' on the school curriculum by the NCERT (National Council of Educational Research and Training).

As has been pointed out earlier on several occasions, the so-called ‘Vedic Mathematics’ is neither ‘Vedic’ nor can it be dignified by the name of mathematics. ‘Vedic Mathematics’,

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as is well-known, originated with a book of the same name by a former Sankaracharya of Puri (the late Jagadguru Swami Shri Bharati Krishna Tirthaji Maharaj) published posthumously in

1965. The book assembled a set of tricks in elementary arithmetic and algebra to be applied in performing computations with numbers and polynomials. As is pointed out even in the foreword to the book by the General Editor, Dr. A.S. Agarwala, the aphorisms in Sanskrit to be found in the book have nothing to do with the Vedas. Nor are these aphorisms to be found in the genuine Vedic literature.

The term “Vedic Mathematics” is therefore entirely misleading and factually incorrect. Further, it is clear from the notation used in the arithmetical tricks in the book that the methods used in this text have nothing to do with the arithmetical techniques of antiquity. Many of the Sanskrit aphorisms in the book are totally cryptic (ancient Indian mathematical writing was anything but cryptic) and often so generalize to be devoid of any specific mathematical meaning. There are several authoritative texts on the mathematics of Vedic times that could be used in part to teach an authoritative and correct account of ancient Indian mathematics but this book clearly cannot be used for any such purpose. The teaching of mathematics involves both the teaching of the basic concepts of the subject as well as methods of mathematical computation. The so-called “Vedic Mathematics” is entirely inadequate to this task considering that it is largely made up of tricks to do some elementary arithmetic computations. Many of these can be far more easily performed on a simple computer or even an advanced calculator.

The book “Vedic Mathematics” essentially deals with arithmetic of the middle and high-school level. Its claims that

“there is no part of mathematics, pure or applied, which is beyond their jurisdiction” is simply ridiculous. In an era when the content of mathematics teaching has to be carefully designed to keep pace with the general explosion of knowledge and the needs of other modern professions that use mathematical techniques, the imposition of “Vedic Mathematics” will be nothing short of calamitous.

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India today has active and excellent schools of research and teaching in mathematics that are at the forefront of modern research in their discipline with some of them recognised as being among the best in the world in their fields of research. It is noteworthy that they have cherished the legacy of distinguished Indian mathematicians like Srinivasa Ramanujam, V. K. Patodi, S. Minakshisundaram, Harish Chandra, K. G. Ramanathan, Hansraj Gupta, Syamdas Mukhopadhyay, Ganesh Prasad, and many others including several living Indian mathematicians. But not one of these schools has lent an iota of legitimacy to ‘Vedic Mathematics’. Nowhere in the world does any school system teach “Vedic Mathematics” or any form of ancient mathematics for that matter as an adjunct to modern mathematical teaching. The bulk of such teaching belongs properly to the teaching of history and in particular the teaching of the history of the sciences.

We consider the imposition of ‘Vedic Mathematics’ by a Government agency, as the perpetration of a fraud on our children, condemning particularly those dependent on public education to a sub-standard mathematical education. Even if we assumed that those who sought to impose ‘Vedic Mathematics’ did so in good faith, it would have been appropriate that the NCERT seek the assistance of renowned Indian mathematicians to evaluate so-called “Vedic Mathematics” before making it part of the National Curricular framework for School Education. Appallingly they have not done so. In this context we demand that the NCERT submit the proposal for the introduction of

‘Vedic Mathematics’ in the school curriculum to recognized bodies of mathematical experts in India, in particular the National Board of Higher Mathematics (under the Dept. of Atomic Energy), and the Mathematics sections of the Indian Academy of Sciences and the Indian National Science Academy, for a thorough and critical examination. In the meanwhile no attempt should be made to thrust the subject into the school curriculum either through the centrally administered school system or by trying to impose it on the school systems of various States.

We are concerned that the essential thrust behind the campaign to introduce the so-called ‘Vedic Mathematics’ has

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more to do with promoting a particular brand of religious majoritarianism and associated obscurantist ideas rather than any serious and meaningful development of mathematics teaching in India. We note that similar concerns have been expressed about other aspects too of the National Curricular Framework for School Education. We re-iterate our firm conviction that all teaching and pedagogy, not just the teaching of mathematics, must be founded on rational, scientific and secular principles.

[Many eminent scholars, researchers from renowned Indian foreign universities have signed this. See the end of section for a detailed list.]

We now give the article “Stop this Fraud on our Children!”

from Peoples Democracy.

Over a hundred leading scientists, academicians, teachers and educationists, in a statement have protested against the attempts by the Vajpayee government to introduce Vedic Mathematics and Vedic Astrology courses in the education system. They have in one voice demanded “Stop this Fraud on our Children!”

The scientists and mathematicians are deeply concerned that the essential thrust behind the campaign to introduce the so- called ‘Vedic Mathematics’ in the school curriculum by the NCERT, and ‘Vedic Astrology’ at the university level by the University Grants Commission, has more to do with promoting a particular brand of religious majoritarianism and associated obscurantist ideas than with any serious development of mathematical or scientific teaching in India. In rejecting these attempts, they re-iterate their firm conviction that all teaching and pedagogy must be founded on rational, scientific and secular principles.

Pointing out that the so-called *"Vedic Mathematics" *is neither vedic nor mathematics, they say that the imposition of

‘*Vedic maths’* will condemn particularly those dependent on public education to a sub-standard mathematical education and will be calamitous for them.

“The teaching of mathematics involves both imparting the basic concepts of the subject as well as methods of mathematical computations. The so-called ‘Vedic maths’ is

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entirely inadequate to this task since it is largely made up of tricks to do some elementary arithmetic computations. Its value is at best recreational and its pedagogical use limited", the statement noted. The signatories demanded that the NCERT submit the proposal for the introduction of ‘Vedic maths’ in the school curriculum for a thorough and critical examination to any of the recognised bodies of mathematical experts in India.

Similarly, they assert that while many people may believe in astrology, this is in the realm of belief and is best left as part of personal faith. Acts of faith cannot be confused with the study and practice of science in the public sphere.

Signatories to the statement include award -winning scientists, Fellows of the Indian National Science Academy, the Indian Academy of Sciences, Senior Professors and eminent mathematicians. Prominent among the over 100 scientists who have signed the statement are:

1. Yashpal (Professor, Eminent Space Scientist, Former

Chairman, UGC),

2. J.V.Narlikar (Director, Inter University Centre for Astronomy and Astrophysics, Pune)

3. M.S.Raghunathan (Professor of Eminence, School of Maths, TIFR and Chairman National Board for Higher Maths).

4. S G Dani, (Senior Professor, School of Mathematics, TIFR)

5. R Parthasarathy (Senior Professor, School of Mathematics, TIFR),

6. Alladi Sitaram (Professor, Indian Statistical Institute (ISI), Bangalore),

7. Vishwambar Pati (Professor, Indian Statistical Institute , Bangalore),

8. Kapil Paranjape (Professor, Institute of Mathematical Sciences

(IMSc), Chennai),

9. S Balachandra Rao, (Principal and Professor of Maths, National College, Bangalore)

10. A P Balachandran, (Professor, Dept. of Physics, Syracuse

University USA),

11. Indranil Biswas (Professor, School of Maths, TIFR)

12. C Musili (Professor, Dept. of Maths and Statistics, Univ. of

Hyderabad),

13. V.S.Borkar (Prof., School of Tech. and Computer Sci., TIFR)

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14. Madhav Deshpande (Prof. of Sanskrit and Linguistics, Dept. of

Asian Languages and Culture, Univ. of Michigan, USA),

15. N. D. Haridass (Senior Professor, Institute of Mathematical

Science, Chennai),

16. V.S. Sunder (Professor, Institute of Mathematical Sciences, Chennai),

17. Nitin Nitsure (Professor, School of Maths, TIFR),

18. T Jayaraman (Professor, Institute of Mathematical Sciences, Chennai),

19. Vikram Mehta (Professor, School of Maths, TIFR),

20. R. Parimala (Senior Professor, School of Maths, TIFR),

21. Rajat Tandon (Professor and Head, Dept. of Maths and

Statistics, Univ. of Hyderabad),

22. Jayashree Ramdas (Senior Reseacrh Scientist, Homi Bhabha

Centre for Science Education, TIFR) ,

23. Ramakrishna Ramaswamy (Professor, School of Physical

Sciences, JNU), D P Sengupta (Retd. Prof. IISc., Bangalore),

24. V Vasanthi Devi (Former VC, Manonmaniam Sundaranar

Univ. Tirunelveli),

25. J K Verma (Professor, Dept. of Maths, IIT Bombay),

26. Bhanu Pratap Das (Professor, Indian Institute of Astrophysics, Bangalore)

27. Pravin Fatnani (Head, Accelerator Controls Centre, Centre for

Advanced Technology, Indore),

28. S.L. Yadava (Professor, TIFR Centre, IISc, Bangalore) ,

29. Kumaresan, S (Professor, Dept. of Mathematics, Univ. of

Mumbai),

30. Rahul Roy (Professor, ISI ,Delhi)

and others….

**2.3**** Views about the B****ook**** in Favour and Against**

The view of his Disciple Manjula Trivedi, Honorary General

Secretary, Sri Vishwa Punarnirmana Sangha, Nagpur written on

16th March 1965 and published in a reprint and revised edition of the book on Vedic Mathematics reads as follows.

“I now proceed to give a short account of the genesis of the work published here. Revered Guruji used to say that he had reconstructed the sixteen mathematical formulae (given in this text) from the Atharveda after assiduous research and ‘Tapas’

55

for about eight years in the forests surrounding Sringeri. *Obviously these formulae are not to be found in the present recensions of Atharvaveda; they were actually reconstructed, on the basis of intuitive revelation, from materials scattered here and there in the Atharvaveda. *Revered Gurudeva used to say that he had written sixteen volumes on these sutras one for each sutra and that the manuscripts of the said volumes were deposited at the house of one of his disciples. Unfortunately the said manuscripts were lost irretrievably from the place of their deposit and this colossal loss was finally confirmed in 1956.

Revered Gurudeva was not much perturbed over this irretrievable loss and used to say that everything was there in his memory and that he would rewrite the 16 volumes!

In 1957, when he had decided finally to undertake a tour of the USA he rewrote from memory the present volume giving an introductory account of the sixteen formulae reconstructed by him …. The present volume is the only work on mathematics that has been left over by Revered Guruji.

The typescript of the present volume was left over by Revered Gurudeva in USA in 1958 for publication. He had been given to understand that he would have to go to the USA for correction of proofs and personal supervision of printing. But his health deteriorated after his return to India and finally the typescript was brought back from the USA after his attainment of Mahasamadhi in 1960.”

A brief sketch from the Statesman, India dated 10th Jan

1956 read as follows. “Sri Shankaracharya denies any spiritual or miraculous powers giving the credit for his revolutionary knowledge to anonymous ancients, who in 16 sutras and 120 words laid down simple formulae for all the world’s mathematical problems […]. I could read a short descriptive note he had prepared on, “The Astounding Wonders of Ancient Indian Vedic Mathematics”. His Holiness, it appears, had spent years in contemplation, and while going through the Vedas had suddenly happened upon the key to what many historians, devotees and translators had dismissed as meaningless jargon. There, contained in certain Sutras, were the processes of mathematics, psychology, ethics and metaphysics.

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“During the reign of King Kamsa” read a sutra, “rebellions, arson, famines and insanitary conditions prevailed”. Decoded this little piece of libelous history gave decimal answer to the fraction 1/17, sixteen processes of simple mathematics reduced to one.

The discovery of one key led to another, and His Holiness found himself turning more and more to the astounding knowledge contained in words whose real meaning had been lost to humanity for generations. This loss is obviously one of the greatest mankind has suffered and I suspect, resulted from the secret being entrusted to people like myself, to whom a square root is one of life’s perpetual mysteries. Had it survived, every – educated ‘soul’ would be a mathematical ‘wizard’ and maths ‘masters’ would “starve”. For my note reads “Little children merely look at the sums written on the blackboard and immediately shout out the answers they have … [Pages 353-355

Vedic Mathematics]

We now briefly quote the views of S.C. Sharma, Ex Head of the Department of Mathematics, NCERT given in Mathematics Today, September 1986.

“The epoch-making and monumental work on Vedic Mathematics unfolds a new method of approach. It relates to the truth of numbers and magnitudes equally applicable to all sciences and arts.

The book brings to light how great and true knowledge is born of intuition, quite different from modern western method. The ancient Indian method and its secret techniques are examined and shown to be capable of solving various problems of mathematics. The universe we live in has a basic mathematical structure obeying the rules of mathematical measures and relations. All the subjects in mathematics – Multiplication, Division, Factorization Equations of calculus Analytical Conics etc. are dealt with in forty chapters vividly working out all problems, in the easiest ever method discovered so far. The volume more a magic is the result of institutional visualization of fundamental mathematical truths born after eight years of highly concentrated endeavor of Jagadguru Sri Bharati Krishna Tirtha.

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Throughout this book efforts have been made to solve the problems in a short time and in short space also …, one can see that the formulae given by the author from Vedas are very interesting and encourage a young mind for learning mathematics as it will not be a bugbear to him”.

This writing finds its place in the back cover of the book of Vedic Mathematics of Jagadguru. Now we give the views of Bibek Debroy, “The fundamentals of Vedic Mathematics” pp.

126-127 of Vedic Mathematics in Tamil volume II).

“Though Vedic Mathematics evokes Hindutva connotations, the fact is, it is a system of simple arithmetic, which can be used for intricate calculations.

The resurgence of interest in Vedic Mathematics came about as a result of Jagadguru Swami Sri Bharati Krishna Tirthaji Maharaj publishing a book on the subject in 1965. Then recently the erstwhile Bharatiya Janata Party governments in Uttar Pradesh, Madhya Pradesh and Himachal Pradesh introduced Vedic Mathematics into the school syllabus, but this move was perceived as an attempt to impose Hindutva, because Vedic philosophy was being projected as the repository of all human wisdom. The subsequent hue and cry over the teaching of Vedic Mathematics is mainly because it has come to be identified with, fundamentalism and obscurantism, both considered poles opposite of science. The critics argue that belief in Vedic Mathematics automatically necessitates belief in Hindu renaissance. But Tirtha is not without his critics, even apart from those who consider Vedic maths is “unscientific”.

**2.4**** Vedas: Repositories of Ancient Indian Lore**

Extent texts of the Vedas do not contain mathematical formulae but they have been found in later associated works. Jagadguru the author of Vedic Mathematics says he has discovered 16 mathematical formulae, …

A standard criticism is that the Vedic Mathematics text is limited to middle and high school formulations and the emphasis is on a series of problem solving tricks. The critics also point out that the Atharva Veda appendix containing

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Tirtha’s 16 mathematical formulae, is not to be found in any part of the existing texts. A third criticism is the most pertinent. The book is badly written. (p.127, Vedic Mathematics 2) [85]. We shall now quote the preface given by His Excellency Dr. L.M.Singhvi, High Commissioner for India in UK, given in pp. V to VI Reprint Vedic Mathematics 2005, Book 2, [51].

Vedic Mathematics for schools is an exceptional book. It is not only a sophisticated pedagogic tool but also an introduction to an ancient civilization. It takes us back to many millennia of India’s mathematical heritage…

The real contribution of this book, “Vedic Mathematics for schools, is to demonstrate that Vedic Mathematics belongs not only to an hoary antiquity but is any day as modern as the day after tomorrow. What distinguishes it particularly is that it has been fashioned by British teachers for use at St.James independent schools in London and other British schools and that it takes its inspiration from the pioneering work of the late Sankaracharya of Puri…

Vedic Mathematics was traditionally taught through aphorisms or Sutras. A sutra is a thread of knowledge, a theorem, a ground norm, a repository of proof. It is formulated as a proposition to encapsulate a rule or a principle. Both Vedic Mathematics and Sanskrit grammar built on the foundations of rigorous logic and on a deep understanding of how the human mind works. The methodology of Vedic Mathematics and of Sanskrit grammar help to hone the human intellect and to guide and groom the human mind into modes of logical reasoning.”

**2.5**** A R****a****tional**** Approach to Study Ancient Literature**

Excerpted from Current Science Vol. 87, No. 4, 25 Aug. 2004.

It was interesting to read about Hertzstark’s hand-held mechanical calculator, which converted subtraction into addition. But I would like to comment on the ‘Vedic Mathematics’ referred to in the note. Bharati Krishna Tirtha is a good mathematician, but the term ‘Vedic Mathematics’ coined by him is misleading, because his mathematics has nothing to do with the *Vedas*. It is his 20th century invention, which should

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be called ‘rapid mathematics’ or ‘Shighra Ganita’. He has disguised his intention of giving it an aura of discovering ancient knowledge with the following admission in the foreword of his book, which few people take the trouble to read. He says there that he saw (thought of) of his Sutras just like the Vedic Rishis saw (thought of) the Richas. That is why he has called his method ‘Vedic Mathematics’. This has made it attractive to the ignorant and not-so ignorant public. I hope scientists will take note of this fact. Vedic astrology is another term, which fascinates people and captures their imagination about its ancient origin. Actually, there is no mention of horoscope and planetary influence in Vedic literature. It only talks of Tithis and Nakshatras as astronomical entities useful for devising a calendar controlled by a series of sacrifices. Astrology of planets originated in Babylon, where astronomers made regular observations of planets, but could not understand their complicated motions. Astrology spread from there to Greece and Europe in the west and to India in the east. There is nothing Vedic about it. It appears that some Indian intellectuals would use the word Vedic as a brand name to sell their ideas to the public. It is imperative that scientists should study ancient literature from a rational point of view, consistent with the then contemporary knowledge.”

**2.6**** Shanghai R****a****nkings**** and Indian Universities**

This article is from Current Science Vol. 87, No. 4, 25

August 2004 [7].

“The editorial “The Shanghai Ranking” is a shocking revelation about the fate of higher education and a slide down of scientific research in India. None of the reputed '5 star' Indian universities qualifies to find a slot among the top 500 at the global level. IISc Bangalore and IITs at Delhi and Kharagpur provide some redeeming feature and put India on the score board with a rank between 250 and 500. Some of the interesting features of the Shanghai rankings are noteworthy: (i) Among the top 99 in the world, we have universities from USA (58), Europe (29), Canada (4), Japan (5), Australia (2) and Israel (1). (ii) On the

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Asia-Pacific list of top 90, we have maximum number of universities from Japan (35), followed by China (18) including Taiwan (5) and Hongkong (5), Australia (13), South Korea (8), Israel (6), India (3), New Zealand (3), Singapore (2) and Turkey

(2). (iii) Indian universities lag behind even small Asian countries, viz. South Korea, Israel, Taiwan and Hongkong, in ranking. I agree with the remark, ‘Sadly, the real universities in India are limping, with the faculty disinterested in research outnumbering those with an academic bent of mind’. The malaise is deep rooted and needs a complete overhaul of the Indian education system.”

**2.7**** ****Conclusions**** ****derived**** ****on**** ****Vedic**** ****Mathematics**** ****an****d**** the**

**Calculations**** of Guru Tirthaji - Secrets of Ancient Maths **This article was translated and revised by its author Jan Hogendijk from his original version published in Dutch in the *Nieuwe Wiskrant* vol. 23 no.3 (March 2004), pp. 49–52.

“The “Vedic” methods of mental calculations in the decimal system are all based on the book Vedic Mathematics by Jagadguru (world guru) Swami (monk) Sri (reverend) Bharati Krsna Tirthaji Maharaja, which appeared in 1965 and which has been reprinted many times [51].

The book contains sixteen brief sutras that can be used for mental calculations in the decimal place-value system. An example is the sutra Ekadhikena Purvena, meaning: by one more than the previous one. The Guru explains that this sutra can for example be used in the mental computation of the period of a recurring decimal fraction such as 1/19 =

0.052631578947368421. as follows:

The word “Vedic” in the title of the book suggests that these calculations are authentic Vedic Mathematics. The question now arises how the Vedic mathematicians were able to write the recurrent decimal fraction of 1/19, while decimal fractions were unknown in India before the seventeenth century. We will first investigate the origin of the sixteen sutras. We cite the Guru himself [51]:

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“And the contemptuous or, at best, patronizing attitude adopted by some so-called orientalists, indologists, antiquarians, research-scholars etc. who condemned, or light heartedly, nay irresponsibly, frivolously and flippantly dismissed, several abstruse-looking and recondite parts of the Vedas as ‘sheer nonsense’ or as ‘infant-humanity’s prattle,’ and so on … further confirmed and strengthened our resolute determination to unravel the too-long hidden mysteries of philosophy and science contained in ancient India’s Vedic lore, with the consequence that, after eight years of concentrated contemplation in forest-solitude, we were at long last able to recover the long lost keys which alone could unlock the portals thereof.

“And we were agreeably astonished and intensely gratified to find that exceedingly tough mathematical problems (which the mathematically most advanced present day Western scientific world had spent huge lots of time, energy and money on and which even now it solves with the utmost difficulty and after vast labour involving large numbers of difficult, tedious and cumbersome ‘steps’ of working) can be easily and readily solved with the help of these ultra-easy Vedic Sutras (or mathematical aphorisms) contained in the Parisısta (the Appendix-portion) of the Atharvaveda in a few simple steps and by methods which can be conscientiously described as mere ‘mental arithmetic.’ ”

Concerning the applicability of the sixteen sutras to all mathematics, we can consult the Foreword to Vedic Mathematics written by Swami Pratyagatmananda Saraswati. This Swami states that one of the sixteen sutras reads Calanakalana, which can be translated as Becoming. The Guru himself translates the sutra in question as “differential calculus”[4, p. 186]. Using this “translation” the sutra indeed promises applicability to a large area in mathematics; but the sutra is of no help in differentiating or integrating a given function such as f(x) =1/sin x.

Sceptics have tried to locate the sutras in the extant Parisista’s (appendices) of the Atharva-Veda, one of the four Vedas. However, the sutras have never been found in authentic texts of the Vedic period. It turns out that the Guru had “seen” the sutras by himself, just as the authentic Vedas were,

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according to tradition, “seen” by the great Rishi’s or seers of ancient India. The Guru told his devotees that he had “re- constructed” his sixteen sutras from the Atharva-Veda in the eight years in which he lived in the forest and spent his time on contemplation and ascetic practices. The book Vedic Mathematics is introduced by a General Editor’s Note [51], in which the following is stated about the sixteen sutras: “[the] style of language also points to their discovery by Sri Swamiji

(the Guru)himself.”

Now we know enough about the authentic Katapayadi system to identify the origin of the Guru’s verse about π / 10. Here is the verse: (it should be noted that the abbreviation r represents a vowel in Sanskrit):

gopi bhagya madhuvrata srngiso dadhi sandhiga Khala jivita Khatava Gala hala rasandhara.

According to the guru, decoding the verse produces the following number:

31415 92653 58979 32384 62643 38327 92

In this number we recognize the first 31 decimals of π (the

32th decimal of π is 5). In the authentic Katapayadi system, the decimals are encoded in reverse order. So according to the authentic system, the verse is decoded as

29723 83346 26483 23979 85356 29514 13

We conclude that the verse is not medieval, and certainly not Vedic. In all likelihood, the guru is the author of the verse.

There is nothing intrinsically wrong with easy methods of mental calculations and mnemonic verses for π. However, it was a miscalculation on the part of the Guru to present his work as ancient Vedic lore. Many experts in India know that the relations between the Guru’s methods and the Vedas are faked. In 1991 the supposed “Vedic” methods of mental calculation

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were introduced in schools in some cities, perhaps in the context of the political program of saffronisation, which emphasizes Hindu religious elements in society (named after the saffron garments of Hindu Swamis). After many protests, the “Vedic” methods were omitted from the programs, only to be reintroduced a few years later. In 2001, a group of intellectuals in India published a statement against the introduction of the Guru’s “Vedic” mathematics in primary schools in India.

Of course, there are plenty of real highlights in the ancient and medieval mathematical tradition of India. Examples are the real Vedic sutras that we have quoted in the beginning of this paper; the decimal place-value system for integers; the concept of sine; the cyclic method for finding integer solutions x, y of

2 2

the “equation of Pell” in the form px + 1 = y (for pa given integer); approximation methods for the sine and arctangents equivalent to modern Taylor series expansions; and so on. Compared to these genuine contributions, the Guru’s mental calculation are of very little interest. In the same way, the Indian philosophical tradition has a very high intrinsic value, which does not need to be “proved” by the so-called applications invented by Guru Tirthaji.

**References**

[1] Chandra Hari, K., 1999: A critical study of Vedic mathematics of Sankaracharya Sri Bharati Krsna Tirthaji Maharaj. Indian Journal of History of Science, 34, 1–17.

[2] Gold, D. and D. Pingree, 1991: A hitherto unknown Sanskrit work concerning Madhava’s derivation of the power series for sine and cosine. Historia Scientiarum, 42, 49–65.

[3] Gupta, R. C., 1994: Six types of Vedic Mathematics. Ganita

Bharati 16, 5–15.

[4] Jagadguru Swami Sri Bharati Krsna Tirthaji Maharaja, 1992: Vedic Mathematics. Delhi: Motilal Banarsidas, revised edition.

[5] Sen, S. N. and A. K. Bag, 1983: The Sulbasutras. New

Delhi: Indian National Science Academy.

[6] Interesting web site on Vedic ritual: http://www.jyoti stoma.nl.

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