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1.
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 October and 5 November 1993)

We in India have good reasons to be proud ing an awareness of our past achievements, on

of a rich heritage in science, philosophy and cul- the strength of authentic information, a more ur-

ture in general, coming to us down the ages. In gent need has also arisen to confront and ex-

mathematics, which is my own area of special- pose such baseless constructs before it is too

isation, the ancient Indians not only took great late. This is not merely a question of setting the

strides long before the Greek advent, which is record straight. The motivated versions have a

a standard reference point in the Western his- way of corrupting the intellectual processes in

torical perspective, but also enriched it for a society and weakening their very foundations in

long period making in particular some very fun- the long run, which needs to be prevented at all

damental contributions such as the place-value costs.

system for writing numbers as we have today, The so-called ” Vedic mathematics” is a case

introduction of zero and so on. Further, the sus- in point. A book by that name written by Ja-

tained development of mathematics in India in gadguru Swami Shri Bharati Krishna Tirthaji

the post-Greek period was indirectly instrumen- Maharaja (Tirathji, 1965) is at the centre of this

tal in the revival in Europe after ”its dark ages”. pursuit, which has now acquired wide follow-

Notwithstanding the enviable background, ing; Tirthaji was the Shankaracharya of Govard-

lack of adequate attention to academic pursuits han Math, Puri, from 1925 until he passed away

over a prolonged period, occasioned by sev- in 1960. The book was published posthumously,

eral factors, together with about two centuries but he had been carrying out a campaign on

of Macaulayan educational system, has unfor- the theme for a long time, apparently for sev-

tunately resulted, on the one hand, in a lack of eral decades, by means of lectures, blackboard

awareness of our historical role in actual terms demonstrations, classes and so on. It has been

and, on the other, an empty sense of pride which known from the beginning that there is no evi-

is more of an emotional reaction to the colonial dence of the contents of the book being of Vedic

domination rather than an intellectual challenge. origin; the Foreword to the book by the General

Together they provide a convenient ground for Editor, Dr. A.S.Agrawala, and an account of the

extremist and misguided elements in society to genesis of the work written by Manjula Trivedi,

”reconstruct history” from nonexistent or con- a disciple of the swamiji, make this clear even

cocted source material to whip up popular eu- before one gets to the text of the book. No one

phoria. has come up with any positive evidence subse-

That this anti-intellectual endeavour is quently either.

counter-productive in the long run and, more im- There has, however, been a persistent pro-

portant, harmful to our image as a mature soci- paganda that the material is from the Vedas. In

ety, is either not recognised or ignored in favour the face of a false sense of national pride associ-

of short-term considerations. Along with the ob- ated with it and the neglect, on the part of the

vious need to accelerate the process of creat- knowledgeable, in countering the propaganda,

1

S.G. Dani

School of Mathematics

Tata Institute of Fundamental Research

(An updated version of the 2-part article in Frontline, 22 October and 5 November 1993)

We in India have good reasons to be proud ing an awareness of our past achievements, on

of a rich heritage in science, philosophy and cul- the strength of authentic information, a more ur-

ture in general, coming to us down the ages. In gent need has also arisen to confront and ex-

mathematics, which is my own area of special- pose such baseless constructs before it is too

isation, the ancient Indians not only took great late. This is not merely a question of setting the

strides long before the Greek advent, which is record straight. The motivated versions have a

a standard reference point in the Western his- way of corrupting the intellectual processes in

torical perspective, but also enriched it for a society and weakening their very foundations in

long period making in particular some very fun- the long run, which needs to be prevented at all

damental contributions such as the place-value costs.

system for writing numbers as we have today, The so-called ” Vedic mathematics” is a case

introduction of zero and so on. Further, the sus- in point. A book by that name written by Ja-

tained development of mathematics in India in gadguru Swami Shri Bharati Krishna Tirthaji

the post-Greek period was indirectly instrumen- Maharaja (Tirathji, 1965) is at the centre of this

tal in the revival in Europe after ”its dark ages”. pursuit, which has now acquired wide follow-

Notwithstanding the enviable background, ing; Tirthaji was the Shankaracharya of Govard-

lack of adequate attention to academic pursuits han Math, Puri, from 1925 until he passed away

over a prolonged period, occasioned by sev- in 1960. The book was published posthumously,

eral factors, together with about two centuries but he had been carrying out a campaign on

of Macaulayan educational system, has unfor- the theme for a long time, apparently for sev-

tunately resulted, on the one hand, in a lack of eral decades, by means of lectures, blackboard

awareness of our historical role in actual terms demonstrations, classes and so on. It has been

and, on the other, an empty sense of pride which known from the beginning that there is no evi-

is more of an emotional reaction to the colonial dence of the contents of the book being of Vedic

domination rather than an intellectual challenge. origin; the Foreword to the book by the General

Together they provide a convenient ground for Editor, Dr. A.S.Agrawala, and an account of the

extremist and misguided elements in society to genesis of the work written by Manjula Trivedi,

”reconstruct history” from nonexistent or con- a disciple of the swamiji, make this clear even

cocted source material to whip up popular eu- before one gets to the text of the book. No one

phoria. has come up with any positive evidence subse-

That this anti-intellectual endeavour is quently either.

counter-productive in the long run and, more im- There has, however, been a persistent pro-

portant, harmful to our image as a mature soci- paganda that the material is from the Vedas. In

ety, is either not recognised or ignored in favour the face of a false sense of national pride associ-

of short-term considerations. Along with the ob- ated with it and the neglect, on the part of the

vious need to accelerate the process of creat- knowledgeable, in countering the propaganda,

1

2.
even educated and well meaning people have while edifice in mathematics (as also in many

tended to accept it uncritically. The vested in- other areas). Harish Chandra’s work is held

terests have also involved politicians in the pro- in great esteem all over the world and sev-

paganda process to gain state support. Several eral leading seats of learning of our times

leaders have lent support to the ”Vedic mathe- pride themselves in having members pursu-

matics” over the years, evidently in the belief ing his ideas; (see, for instance, Langlands,

of its being from ancient scriptures. In the cur- 1993). Even among those based in India, several

rent environment, when a label as ancient seems like Syamdas Mukhopadhyay, Ganesh Prasad,

to carry considerable premium irrespective of its B.N.Prasad, K.Anand Rau, T.Vijayaraghavan,

authenticity or merit, the purveyors would have S.S.Pillai, S.Minakshisundaram, Hansraj

it going easy. Gupta, K.G.Ramanathan, B.S.Madhava Rao,

Large sums have been spent both by the V.V.Narlikar, P.L.Bhatnagar and so on and also

Government and several private agencies to sup- many living Indian mathematicians have carved

port this ”Vedic mathematics”, while authentic a niche for themselves on the international math-

Vedic studies continue to be neglected. People, ematical scene (see Narasimhan, 1991). Ignor-

especially children, are encouraged to learn and ing all this while introducing the swamiji’s name

spread the contents of the book, largely on the in the ”brief history” would inevitably create a

baseless premise of their being from the Vedas. warped perspective in children’s minds, favour-

With missionary zeal several ”devotees” of this ing gimmickry rather than professional work.

cause have striven to take the ”message” around What does the swamiji’s ”Vedic mathematics”

the world; not surprisingly, they have even met seek to do and what does it achieve? In his pref-

with some success in the West, not unlike some ace of the book, grandly titled ” A Descriptive

of the gurus and yogis peddling their own ver- Prefatory Note on the astounding Wonders of

sions of ”Indian philosophy”. Several people Ancient Indian Vedic Mathematics,” the swamiji

are also engaged in ”research” in the new ”Vedic tells us that he strove from his childhood to

mathematics.” study the Vedas critically ” to prove to ourselves

To top it all, when in the early nineties the (and to others) the correctness (or otherwise)”of

Uttar Pradesh Government introduced ”Vedic the ”derivational meaning” of ”Veda” that the

mathematics” in school text books, the contents ” Vedas should contain within themselves all

of the swamiji’s book were treated as if they the knowledge needed by the mankind relating

were genuinely from the Vedas; this also nat- not only to spiritual matters but also those usu-

urally seems to have led them to include a list ally described as purely ’secular’, ’temporal’ or

of the swamiji’s sutras on one of the opening ’worldly’; in other words, simply because of the

pages (presumably for the students to learn them meaning of the word ’Veda’, everything that is

by heart and recite!) and to accord the swamiji worth knowing is expected to be contained in

a place of honour in the ” brief history of In- the vedas and the swamiji seeks to prove it to be

dian mathematics” described in the beginning of the case!

the textbook, together with a chart, which cu- It may be worthwhile to point out here that

riously has Srinivasa Ramanujan’s as the only there would be room for starting such an enter-

other name from the twentieth century! prise with the word ’science’! He also describes

For all their concern to inculcate a sense how the ” contemptuous or at best patronising

of national pride in children, those respon- ” attitude of Orientalists, Indologists and so on

sible for this have not cared for the simple strengthened his determination to unravel the

fact that modern India has also produced sev- too-long-hidden mysteries of philosophy and

eral notable mathematicians and built a worth- science contained in ancient India’s Vedic lore,

2

tended to accept it uncritically. The vested in- other areas). Harish Chandra’s work is held

terests have also involved politicians in the pro- in great esteem all over the world and sev-

paganda process to gain state support. Several eral leading seats of learning of our times

leaders have lent support to the ”Vedic mathe- pride themselves in having members pursu-

matics” over the years, evidently in the belief ing his ideas; (see, for instance, Langlands,

of its being from ancient scriptures. In the cur- 1993). Even among those based in India, several

rent environment, when a label as ancient seems like Syamdas Mukhopadhyay, Ganesh Prasad,

to carry considerable premium irrespective of its B.N.Prasad, K.Anand Rau, T.Vijayaraghavan,

authenticity or merit, the purveyors would have S.S.Pillai, S.Minakshisundaram, Hansraj

it going easy. Gupta, K.G.Ramanathan, B.S.Madhava Rao,

Large sums have been spent both by the V.V.Narlikar, P.L.Bhatnagar and so on and also

Government and several private agencies to sup- many living Indian mathematicians have carved

port this ”Vedic mathematics”, while authentic a niche for themselves on the international math-

Vedic studies continue to be neglected. People, ematical scene (see Narasimhan, 1991). Ignor-

especially children, are encouraged to learn and ing all this while introducing the swamiji’s name

spread the contents of the book, largely on the in the ”brief history” would inevitably create a

baseless premise of their being from the Vedas. warped perspective in children’s minds, favour-

With missionary zeal several ”devotees” of this ing gimmickry rather than professional work.

cause have striven to take the ”message” around What does the swamiji’s ”Vedic mathematics”

the world; not surprisingly, they have even met seek to do and what does it achieve? In his pref-

with some success in the West, not unlike some ace of the book, grandly titled ” A Descriptive

of the gurus and yogis peddling their own ver- Prefatory Note on the astounding Wonders of

sions of ”Indian philosophy”. Several people Ancient Indian Vedic Mathematics,” the swamiji

are also engaged in ”research” in the new ”Vedic tells us that he strove from his childhood to

mathematics.” study the Vedas critically ” to prove to ourselves

To top it all, when in the early nineties the (and to others) the correctness (or otherwise)”of

Uttar Pradesh Government introduced ”Vedic the ”derivational meaning” of ”Veda” that the

mathematics” in school text books, the contents ” Vedas should contain within themselves all

of the swamiji’s book were treated as if they the knowledge needed by the mankind relating

were genuinely from the Vedas; this also nat- not only to spiritual matters but also those usu-

urally seems to have led them to include a list ally described as purely ’secular’, ’temporal’ or

of the swamiji’s sutras on one of the opening ’worldly’; in other words, simply because of the

pages (presumably for the students to learn them meaning of the word ’Veda’, everything that is

by heart and recite!) and to accord the swamiji worth knowing is expected to be contained in

a place of honour in the ” brief history of In- the vedas and the swamiji seeks to prove it to be

dian mathematics” described in the beginning of the case!

the textbook, together with a chart, which cu- It may be worthwhile to point out here that

riously has Srinivasa Ramanujan’s as the only there would be room for starting such an enter-

other name from the twentieth century! prise with the word ’science’! He also describes

For all their concern to inculcate a sense how the ” contemptuous or at best patronising

of national pride in children, those respon- ” attitude of Orientalists, Indologists and so on

sible for this have not cared for the simple strengthened his determination to unravel the

fact that modern India has also produced sev- too-long-hidden mysteries of philosophy and

eral notable mathematicians and built a worth- science contained in ancient India’s Vedic lore,

2

3.
with the consequence that,”after eight years of to do so, but when Prof.K.S.Shukla, a renowned

concentrated contemplation in forest solitude, scholar of ancient Indian mathematics, met him

we were at long last able to recover the long in 1950, when the swamiji visited Lucknow to

lost keys which alone could unlock the portals give a blackboard demonstration of his ”Vedic

thereof.” mathematics”, and requested him to point out

The mindset revealed in this can hardly be the sutras in question in the Parishishta of the

said to be suitable in scientific and objective Atharva Veda, of which he even carried a copy

inquiry or pursuit of knowledge, but perhaps (the standard version edited by G.M.Bolling and

one should not grudge it in someone from a to- J.Von Negelein), the swamiji is said to have

tally different milieu, if the outcome is positive. told him that the 16 sutra demonstrated by him

One would have thought that with all the com- were not in those Parishishtas and that ”they oc-

mitment and grit the author would have come curred in his own Parishishta and not any other”

up with at least a few new things which can (Shukla, 1980, or Shukla, 1991). What justifica-

be attributed to the Vedas, with solid evidence. tion the swamiji thought he had for introducing

This would have made a worthwhile contribu- an appendix in the Atharva Veda, the contents

tion to our understanding of our heritage. In- of which are nevertheless to be viewed as from

stead, all said and done there is only the author’s the Veda, is anybody’s guess. In any case, even

certificate that ”we were agreeably astonished such a Parishishta, written by the swamiji, does

and intensely gratified to find that exceedingly not exist in the form of a Sanskrit text.

though mathematical problems can be easily and Let us suppose for a moment that the author

readily solved with the help of these ultra-easy indeed found the sutras in some manuscript of

Vedic sutras (or mathematical aphorisms) con- the Atharva Veda, which he came across. Would

tained in the Parishishta (the appendix portion) he not then have preserved the manuscript?

of the Atharva Veda in a few simple steps and Would he not have shown at least to some people

by methods which can be conscientiously de- where the sutras are in the manuscript? Would

scribed as mere ’mental arithmetic’ (paragraph he not have revealed to some cherished students

9 in the preface). That passing reference to the how to look for sutras with such profound math-

Atharva Veda is all that is ever said by way of ematical implications as he attributes to the su-

source material for the contents. The sutras, in- tras in question, in that or other manuscripts

cidentally, which appeared later scattered in the that may be found? While there is a specific

book, are short phrases of just about two to four mention in the write-up of Manjula Trivedi,

words in Sanskrit, such as Ekadhikena Purvena in the beginning of the book, about some 16-

or Anurupye Shunyam Anyat. (There are 16 of volume manuscript written by the swamiji hav-

them and in addition there are 13 of what are ing been lost in 1956, there is no mention what-

called sub-sutras, similar in nature to the sutras). ever (let alone any lamentation that would be

The first key question, which would occur to due in such an event) either in her write-up

anyone, is where are these sutras to be found in nor in the swamiji’s preface about any original

the Atharva Veda. One does not mean this as a manuscript having been lost. No one certainly

rhetorical question. Considering that at the out- has come forward with any information received

set the author seemed set to send all doubting from the swamiji with regard to the other ques-

Thomases packing, the least one would expect tions above. It is to be noted that want of time

is that he would point out where the sutras are, could not be a factor in any of this, since the

say in which part, stanza, page and so on, espe- swamiji kindly informs us in the preface that ”

cially since it is not a small article that is being Ever since (i.e. since several decades ago), we

referred to. Not only has the author not cared have been carrying on an incessant and strenu-

3

concentrated contemplation in forest solitude, scholar of ancient Indian mathematics, met him

we were at long last able to recover the long in 1950, when the swamiji visited Lucknow to

lost keys which alone could unlock the portals give a blackboard demonstration of his ”Vedic

thereof.” mathematics”, and requested him to point out

The mindset revealed in this can hardly be the sutras in question in the Parishishta of the

said to be suitable in scientific and objective Atharva Veda, of which he even carried a copy

inquiry or pursuit of knowledge, but perhaps (the standard version edited by G.M.Bolling and

one should not grudge it in someone from a to- J.Von Negelein), the swamiji is said to have

tally different milieu, if the outcome is positive. told him that the 16 sutra demonstrated by him

One would have thought that with all the com- were not in those Parishishtas and that ”they oc-

mitment and grit the author would have come curred in his own Parishishta and not any other”

up with at least a few new things which can (Shukla, 1980, or Shukla, 1991). What justifica-

be attributed to the Vedas, with solid evidence. tion the swamiji thought he had for introducing

This would have made a worthwhile contribu- an appendix in the Atharva Veda, the contents

tion to our understanding of our heritage. In- of which are nevertheless to be viewed as from

stead, all said and done there is only the author’s the Veda, is anybody’s guess. In any case, even

certificate that ”we were agreeably astonished such a Parishishta, written by the swamiji, does

and intensely gratified to find that exceedingly not exist in the form of a Sanskrit text.

though mathematical problems can be easily and Let us suppose for a moment that the author

readily solved with the help of these ultra-easy indeed found the sutras in some manuscript of

Vedic sutras (or mathematical aphorisms) con- the Atharva Veda, which he came across. Would

tained in the Parishishta (the appendix portion) he not then have preserved the manuscript?

of the Atharva Veda in a few simple steps and Would he not have shown at least to some people

by methods which can be conscientiously de- where the sutras are in the manuscript? Would

scribed as mere ’mental arithmetic’ (paragraph he not have revealed to some cherished students

9 in the preface). That passing reference to the how to look for sutras with such profound math-

Atharva Veda is all that is ever said by way of ematical implications as he attributes to the su-

source material for the contents. The sutras, in- tras in question, in that or other manuscripts

cidentally, which appeared later scattered in the that may be found? While there is a specific

book, are short phrases of just about two to four mention in the write-up of Manjula Trivedi,

words in Sanskrit, such as Ekadhikena Purvena in the beginning of the book, about some 16-

or Anurupye Shunyam Anyat. (There are 16 of volume manuscript written by the swamiji hav-

them and in addition there are 13 of what are ing been lost in 1956, there is no mention what-

called sub-sutras, similar in nature to the sutras). ever (let alone any lamentation that would be

The first key question, which would occur to due in such an event) either in her write-up

anyone, is where are these sutras to be found in nor in the swamiji’s preface about any original

the Atharva Veda. One does not mean this as a manuscript having been lost. No one certainly

rhetorical question. Considering that at the out- has come forward with any information received

set the author seemed set to send all doubting from the swamiji with regard to the other ques-

Thomases packing, the least one would expect tions above. It is to be noted that want of time

is that he would point out where the sutras are, could not be a factor in any of this, since the

say in which part, stanza, page and so on, espe- swamiji kindly informs us in the preface that ”

cially since it is not a small article that is being Ever since (i.e. since several decades ago), we

referred to. Not only has the author not cared have been carrying on an incessant and strenu-

3

4.
ous campaign for the India-wide diffusion of all as to whether they mean or yield, in some cog-

this scientific knowledge”. nisable way, what the author claims; in other

The only natural explanation is that there words, we would still need to know whether

was no such manuscript. It has in fact been men- such a source really contains the mathematics

tioned by Agrawala in his general editor’s fore- the swamiji deals with or merely the phrases,

word to the book, and also by Manjula Trivedi in may be in some quite different context. It is in-

the short account of the genesis of the work, in- teresting to consider the swamiji’s sutras in this

cluded in the book together with a biographical light. One of them, for instance, is Ekadhikena

sketch of the swamiji, that the sutras do not ap- Purvena which literally just means” by one more

pear in hitherto known Parishishtas. The general than the previous one.” In chapter I, the swamiji

editor also notes that the style of language of the tells us that it is a sutra for finding the digits in

sutras ”point to their discovery by Shri Swamiji the decimal expansion of numbers such as 1/19,

himself” (emphasis added); the language style and 1/29, where the denominator is a number

being contemporary can be confirmed indepen- with 9 in the unit’s place; he goes on to give

dently from other Sanskrit scholars as well. The a page-long description of the procedure to be

question why then the contents should be con- followed, whose only connection with the su-

sidered” Vedic” apparently did not bother the tra is that it involves, in particular, repeatedly

general editor, as he agreed with the author multiplying by one more than the previous one,

that ”by definition” the Vedas should contain all namely 2, 3 and so on, respectively, the ”pre-

knowledge (never mind whether found in the vious one” being the number before the unit’s

20th century, or perhaps even later)! Manjula place; the full procedure involves a lot more by

Trivedi, the disciple has of course no problem way of arranging the digits which can in no way

with the sutras not being found in the Vedas as be read off from the phrase.

she in fact says that they were actually recon- In Chapter II, we are told that the same sutra

structed by her beloved ” Gurudeva,” on the ba- also means that to find the square of a number

sis of intuitive revelation from material scattered like 25 and 35, (with five in unit’s place) multi-

here and there in the Atharva Veda, after ” assid- ply the number of tens by one more than itself

uous research and ’Tapas’ for about eight years and write 25 ahead of that; like 625, 1,225 and

in the forests surrounding Shringeri.” Isn’t that so on. The phrase Ekanyunena Purvena which

adequate to consider them to be ”Vedic”? Well, means ” by one less than the previous one” is

one can hardly argue with the devout! There is however given to mean something which has

a little problem as to why the Gurudeva him- neither to do with decimal expansions nor with

self did not say so (that the sutras were recon- squaring of numbers but concerns multiplying

structed) rather than referring to them as sutras together two numbers, one of which has 9 in all

contained in the Parishishta of the Atharva Veda, places (like 99,999, so on.)!

but we will have to let it pass. Anyway the fact Allowing oneself such unlimited freedom of

remains that she was aware that they could not interpretation, one can also interpret the same

actually be located in what we lesser mortals three-word phrase to mean also many other

consider to be the Atharva Veda. things not only in mathematics but also in many

The question of the source of the sutras is other subjects such as physics, chemistry, biol-

merely the first that would come to mind, and ogy, economics, sociology and politics. Con-

already on that there is such a muddle. Actually, sider, for instance, the following ” meaning”:

even if the sutras were to be found, say in the the family size may be allowed to grow, at most,

Atharva Veda or some other ancient text, that by one more than the previous one. In this we

still leaves open another fundamental question have the family-planning message of the 1960s;

4

this scientific knowledge”. nisable way, what the author claims; in other

The only natural explanation is that there words, we would still need to know whether

was no such manuscript. It has in fact been men- such a source really contains the mathematics

tioned by Agrawala in his general editor’s fore- the swamiji deals with or merely the phrases,

word to the book, and also by Manjula Trivedi in may be in some quite different context. It is in-

the short account of the genesis of the work, in- teresting to consider the swamiji’s sutras in this

cluded in the book together with a biographical light. One of them, for instance, is Ekadhikena

sketch of the swamiji, that the sutras do not ap- Purvena which literally just means” by one more

pear in hitherto known Parishishtas. The general than the previous one.” In chapter I, the swamiji

editor also notes that the style of language of the tells us that it is a sutra for finding the digits in

sutras ”point to their discovery by Shri Swamiji the decimal expansion of numbers such as 1/19,

himself” (emphasis added); the language style and 1/29, where the denominator is a number

being contemporary can be confirmed indepen- with 9 in the unit’s place; he goes on to give

dently from other Sanskrit scholars as well. The a page-long description of the procedure to be

question why then the contents should be con- followed, whose only connection with the su-

sidered” Vedic” apparently did not bother the tra is that it involves, in particular, repeatedly

general editor, as he agreed with the author multiplying by one more than the previous one,

that ”by definition” the Vedas should contain all namely 2, 3 and so on, respectively, the ”pre-

knowledge (never mind whether found in the vious one” being the number before the unit’s

20th century, or perhaps even later)! Manjula place; the full procedure involves a lot more by

Trivedi, the disciple has of course no problem way of arranging the digits which can in no way

with the sutras not being found in the Vedas as be read off from the phrase.

she in fact says that they were actually recon- In Chapter II, we are told that the same sutra

structed by her beloved ” Gurudeva,” on the ba- also means that to find the square of a number

sis of intuitive revelation from material scattered like 25 and 35, (with five in unit’s place) multi-

here and there in the Atharva Veda, after ” assid- ply the number of tens by one more than itself

uous research and ’Tapas’ for about eight years and write 25 ahead of that; like 625, 1,225 and

in the forests surrounding Shringeri.” Isn’t that so on. The phrase Ekanyunena Purvena which

adequate to consider them to be ”Vedic”? Well, means ” by one less than the previous one” is

one can hardly argue with the devout! There is however given to mean something which has

a little problem as to why the Gurudeva him- neither to do with decimal expansions nor with

self did not say so (that the sutras were recon- squaring of numbers but concerns multiplying

structed) rather than referring to them as sutras together two numbers, one of which has 9 in all

contained in the Parishishta of the Atharva Veda, places (like 99,999, so on.)!

but we will have to let it pass. Anyway the fact Allowing oneself such unlimited freedom of

remains that she was aware that they could not interpretation, one can also interpret the same

actually be located in what we lesser mortals three-word phrase to mean also many other

consider to be the Atharva Veda. things not only in mathematics but also in many

The question of the source of the sutras is other subjects such as physics, chemistry, biol-

merely the first that would come to mind, and ogy, economics, sociology and politics. Con-

already on that there is such a muddle. Actually, sider, for instance, the following ” meaning”:

even if the sutras were to be found, say in the the family size may be allowed to grow, at most,

Atharva Veda or some other ancient text, that by one more than the previous one. In this we

still leaves open another fundamental question have the family-planning message of the 1960s;

4

5.
the ”previous one” being the couple, the pre- a bit of antique finish!

scription is that they should have no more than An analysis of the mathematical contents of

three children. Thus the lal trikon (red triangle) Tirthaji’s book also shows that they cannot be

formula may be seen to be ” from the Atharva from the Vedas. Though unfortunately there is

Veda,” thanks to the swamiji’s novel technique considerable ignorance about the subject, math-

(with just a bit of credit to yours faithfully). If ematics from the Vedas is far from being an

you think the three children norm now outdated, unexplored area. Painstaking efforts have been

there is no need to despair. One can get the two- made for well over a century to study the origi-

children or even the one-child formula also from nal ancient texts from the point of view of under-

the same sutra; count only the man as the ”pre- standing the extent of mathematical knowledge

vious one” (the woman is an outsider joining in in ancient times. For instance, from the study

marriage, isn’t she) and in the growth of the fam- of Vedic Samhitas and Brahamanas it has been

ily either count only the children or include also noted that they had the system of counting pro-

the wife, depending on what suits the desired gressing in multiples of 10 as we have today and

formula! that they considered remarkably large numbers,

Another sutra is Yavadunam, which means even up to 14 digits, unlike other civilizations of

”as much less;” a lifetime may not suffice to those times. From the Vedanga period there is in

write down all the things such a phrase could fact available a significant body of mathematical

”mean,” in the spirit as above. There is even literature in the form of Shulvasutras, from the

a sub-sutra, Vilokanam (observation) and that period between 800 bc and 500 bc, or perhaps

is supposed to mean various mathematical steps even earlier, some of which contain expositions

involving observation! In the same vein one can of various mathematical principles involved in

actually suggest a single sutra adequate not only construction of sacrificial ’vedi’s needed in per-

for all of mathematics but many many subjects: forming ’yajna’s (see, for instance, Sen and Bag

Chintanam (think)! 1983).

It may be argued that there are, after all, Baudhyana Shulvasutra, the earliest of the

ciphers which convey more information than extant Shulvasutras, already contains, for in-

meets the eye. But the meaning in those cases is stance, what is currently known as Pythagoras’

either arrived at from the knowledge of the deci- Theorem (Sen and Bag, 1983, page 78, 1.12).

phering code or deduced in one or other way us- It is the earliest known explicit statement of the

ing various kinds of contexual information. Nei- theorem in the general form (anywhere in the

ther applies in the present case. The sutras in world) and precedes Pythagoras by at least a

the swamiji’s book are in reality mere names for few hundred years. The texts also show a re-

various steps to be followed in various contexts; markable familiarity with many other facts from

the steps themselves had to be known indepen- the so-called Euclidean Geometry and it is clear

dently. In other words, the mathematical step that considerable use was made of these, long

is not arrived at by understanding or interpret- before the Greeks formulated them. The work

ing what are given as sutras; rather, sutras some- of George Thibaut in the last century and that of

what suggestive of the meaning of the steps are A.Burk around the turn of the century brought

attached to them like names. It is like associ- to the attention of the world the significance of

ating the ’sutra’ VIBGYOR to the sequence of the mathematics of the Shulvasutras. It has been

colours in rainbow (which make up the white followed up in this century by both foreign and

light). Usage of words in Sanskrit, a language Indian historians of mathematics. It is this kind

which the popular mind unquestioningly asso- of authentic work, and not some mumbo-jumbo

ciates with the distant past(!), lend the contents that would highlight our rich heritage. I would

5

scription is that they should have no more than An analysis of the mathematical contents of

three children. Thus the lal trikon (red triangle) Tirthaji’s book also shows that they cannot be

formula may be seen to be ” from the Atharva from the Vedas. Though unfortunately there is

Veda,” thanks to the swamiji’s novel technique considerable ignorance about the subject, math-

(with just a bit of credit to yours faithfully). If ematics from the Vedas is far from being an

you think the three children norm now outdated, unexplored area. Painstaking efforts have been

there is no need to despair. One can get the two- made for well over a century to study the origi-

children or even the one-child formula also from nal ancient texts from the point of view of under-

the same sutra; count only the man as the ”pre- standing the extent of mathematical knowledge

vious one” (the woman is an outsider joining in in ancient times. For instance, from the study

marriage, isn’t she) and in the growth of the fam- of Vedic Samhitas and Brahamanas it has been

ily either count only the children or include also noted that they had the system of counting pro-

the wife, depending on what suits the desired gressing in multiples of 10 as we have today and

formula! that they considered remarkably large numbers,

Another sutra is Yavadunam, which means even up to 14 digits, unlike other civilizations of

”as much less;” a lifetime may not suffice to those times. From the Vedanga period there is in

write down all the things such a phrase could fact available a significant body of mathematical

”mean,” in the spirit as above. There is even literature in the form of Shulvasutras, from the

a sub-sutra, Vilokanam (observation) and that period between 800 bc and 500 bc, or perhaps

is supposed to mean various mathematical steps even earlier, some of which contain expositions

involving observation! In the same vein one can of various mathematical principles involved in

actually suggest a single sutra adequate not only construction of sacrificial ’vedi’s needed in per-

for all of mathematics but many many subjects: forming ’yajna’s (see, for instance, Sen and Bag

Chintanam (think)! 1983).

It may be argued that there are, after all, Baudhyana Shulvasutra, the earliest of the

ciphers which convey more information than extant Shulvasutras, already contains, for in-

meets the eye. But the meaning in those cases is stance, what is currently known as Pythagoras’

either arrived at from the knowledge of the deci- Theorem (Sen and Bag, 1983, page 78, 1.12).

phering code or deduced in one or other way us- It is the earliest known explicit statement of the

ing various kinds of contexual information. Nei- theorem in the general form (anywhere in the

ther applies in the present case. The sutras in world) and precedes Pythagoras by at least a

the swamiji’s book are in reality mere names for few hundred years. The texts also show a re-

various steps to be followed in various contexts; markable familiarity with many other facts from

the steps themselves had to be known indepen- the so-called Euclidean Geometry and it is clear

dently. In other words, the mathematical step that considerable use was made of these, long

is not arrived at by understanding or interpret- before the Greeks formulated them. The work

ing what are given as sutras; rather, sutras some- of George Thibaut in the last century and that of

what suggestive of the meaning of the steps are A.Burk around the turn of the century brought

attached to them like names. It is like associ- to the attention of the world the significance of

ating the ’sutra’ VIBGYOR to the sequence of the mathematics of the Shulvasutras. It has been

colours in rainbow (which make up the white followed up in this century by both foreign and

light). Usage of words in Sanskrit, a language Indian historians of mathematics. It is this kind

which the popular mind unquestioningly asso- of authentic work, and not some mumbo-jumbo

ciates with the distant past(!), lend the contents that would highlight our rich heritage. I would

5

6.
strongly recommend to the reader to peruse the ularity in the 17th century following their use

monograph, The Sulbasutras by S.N.Sen and in John Napier’s logarithm tables (see, for in-

A.K.Bag (Sen and Bag, 1983), containing the stance, Boyer, 1968, page 334).

original sutras, their translation and a detailed Similarly, in chapter XXII the swamiji

commentary, which includes a survey of a num- claims to give ” sutras relevant to successive dif-

ber of earlier works on the subject. There are ferentiation, covering the theorems of Leibnitz,

also several books on ancient Indian mathemat- Maclaurin, Taylor, etc. and a lot of other ma-

ics from the Vedic period. terial which is yet to be studied and decided on

The contents of the swamiji’s book have by the great mathematicians of the present-day

practically nothing in common with what is Western world;” it should perhaps be mentioned

known of the mathematics from the Vedic pe- before we proceed that the chapter does not re-

riod or even with the subsequent rich tradi- ally deal with anything of the sort that would

tion of mathematics in India until the advent even remotely justify such a grandiloquent an-

of the modern era; incidentally, the descriptions nouncement, but rather deals with differentia-

of mathematical principles or procedures in an- tion as an operation on polynomials, which is

cient mathematical texts are quite explicit and a very special case reducing it all to elementary

not in terms of cryptic sutras. The very first algebra devoid of the very soul of calculus, as

chapter of the book (as also chapters XXVI to taught even at the college level.

XXVIII) involves the notion of decimal frac- Given the context, we shall leave Leibnitz

tions in an essential way. If the contents are and company alone, but consider the notions of

to be Vedic, there would have had to be a good derivative and successive differentiation. Did

deal of familiarity with decimal fractions, even the notions exist in the Vedic times? While cer-

involving several digits, at that time. It turns out tain elements preliminary to calculus have been

that while the Shulvasutras make extensive use found in the works of Bhaskara II from the 12th

of fractions in the usual form, nowhere is there century and later Indian mathematicians in the

any indication of fractions in decimal form. It pre-calculus era in international mathematics,

is inconceivable that such an important notion such crystallised notions as the derivative or the

would be left out, had it been known, from what integral were not known. Though a case may

are really like users manuals of those times, pro- be made that the developments here would have

duced at different times over a prolonged period. led to the discovery of calculus in India, no his-

Not only the Shulvasutras and the earlier Vedic torians of Indian mathematics would dream of

works, but even the works of mathematicians proposing that they actually had such a notion

such as Aryabhata, Brahmagupta and Bhaskara, as the derivative, let alone successive differenti-

are not found to contain any decimal fractions. ation; the question here is not about performing

Is it possible that none of them had access to the operation on polynomials, but of the con-

some Vedic source that the swamiji could lay his cept. A similar comment applies with regard

hands on (and still not describe it specifically)? to integration, in chapter XXIV. It should also

How far do we have to stretch our credulity? be borne in mind that if calculus were to be

The fact is that the use of decimal frac- known in India in the early times, it would have

tions started only in the 16th century, propa- been acquired by foreigners as well, long before

gated to a large extent by Francois Viete; the it actually came to be discovered, as there was

use of the decimal point (separating the inte- enough interaction between India and the out-

ger and the fractional parts) itself, as a notation side world.

for the decimal representation, began only to- If this is not enough, in Chapter XXXIX we

wards the end of the century and acquired pop- learn that analytic conics has an ” important and

6

monograph, The Sulbasutras by S.N.Sen and in John Napier’s logarithm tables (see, for in-

A.K.Bag (Sen and Bag, 1983), containing the stance, Boyer, 1968, page 334).

original sutras, their translation and a detailed Similarly, in chapter XXII the swamiji

commentary, which includes a survey of a num- claims to give ” sutras relevant to successive dif-

ber of earlier works on the subject. There are ferentiation, covering the theorems of Leibnitz,

also several books on ancient Indian mathemat- Maclaurin, Taylor, etc. and a lot of other ma-

ics from the Vedic period. terial which is yet to be studied and decided on

The contents of the swamiji’s book have by the great mathematicians of the present-day

practically nothing in common with what is Western world;” it should perhaps be mentioned

known of the mathematics from the Vedic pe- before we proceed that the chapter does not re-

riod or even with the subsequent rich tradi- ally deal with anything of the sort that would

tion of mathematics in India until the advent even remotely justify such a grandiloquent an-

of the modern era; incidentally, the descriptions nouncement, but rather deals with differentia-

of mathematical principles or procedures in an- tion as an operation on polynomials, which is

cient mathematical texts are quite explicit and a very special case reducing it all to elementary

not in terms of cryptic sutras. The very first algebra devoid of the very soul of calculus, as

chapter of the book (as also chapters XXVI to taught even at the college level.

XXVIII) involves the notion of decimal frac- Given the context, we shall leave Leibnitz

tions in an essential way. If the contents are and company alone, but consider the notions of

to be Vedic, there would have had to be a good derivative and successive differentiation. Did

deal of familiarity with decimal fractions, even the notions exist in the Vedic times? While cer-

involving several digits, at that time. It turns out tain elements preliminary to calculus have been

that while the Shulvasutras make extensive use found in the works of Bhaskara II from the 12th

of fractions in the usual form, nowhere is there century and later Indian mathematicians in the

any indication of fractions in decimal form. It pre-calculus era in international mathematics,

is inconceivable that such an important notion such crystallised notions as the derivative or the

would be left out, had it been known, from what integral were not known. Though a case may

are really like users manuals of those times, pro- be made that the developments here would have

duced at different times over a prolonged period. led to the discovery of calculus in India, no his-

Not only the Shulvasutras and the earlier Vedic torians of Indian mathematics would dream of

works, but even the works of mathematicians proposing that they actually had such a notion

such as Aryabhata, Brahmagupta and Bhaskara, as the derivative, let alone successive differenti-

are not found to contain any decimal fractions. ation; the question here is not about performing

Is it possible that none of them had access to the operation on polynomials, but of the con-

some Vedic source that the swamiji could lay his cept. A similar comment applies with regard

hands on (and still not describe it specifically)? to integration, in chapter XXIV. It should also

How far do we have to stretch our credulity? be borne in mind that if calculus were to be

The fact is that the use of decimal frac- known in India in the early times, it would have

tions started only in the 16th century, propa- been acquired by foreigners as well, long before

gated to a large extent by Francois Viete; the it actually came to be discovered, as there was

use of the decimal point (separating the inte- enough interaction between India and the out-

ger and the fractional parts) itself, as a notation side world.

for the decimal representation, began only to- If this is not enough, in Chapter XXXIX we

wards the end of the century and acquired pop- learn that analytic conics has an ” important and

6

7.
predominating place for itself in the Vedic sys- perimenting with numbers might be expected to

tem of mathematics,” and in Chapter XL we come up with. The tricks are, however, based

find a whole list of subjects such as dynam- on well-understood mathematical principles and

ics, statics, hydrostatics, pneumatics and applied there is no mystery about them.

mathematics listed alongside such elementary Of course to produce such a body of tricks,

things as subtractions, ratios, proportions and even using the well-known is still a non-trivial

such money matters as interest and annuities (!), task and there is a serious question of how this

discounts (!) to which we are assured, without came to be accomplished. It is sometimes sug-

going into details, that the Vedic sutras can be gested that Tirthaji himself might have invented

applied. Need we comment any further on this? the tricks. The fact that he had a M.A.degree in

The remaining chapters are mostly elementary mathematics is notable in this context. It is also

in content, on account of which one does not possible that he might have learnt some of the

see such marked incongruities in their respect. tricks from some elders during an early period

It has, however, been pointed out by Shukla that in his life and developed on them during those

many of the topics considered in the book are ”eight years of concentrated contemplation in

alien to the pursuits of ancient Indian mathe- forest solitude:” this would mean that they do in-

maticians, not only form the Vedic period but volve a certain element of tradition, though not

until much later (Shukla, 1989 or Shukla, 1991). to the absurd extent that is claimed. These can,

These include many such topics as factorisation however, be viewed only as possibilities and it

of algebraic expressions, HCF (highest com- would not be easy to settle these details. But

mon factor) of algebraic expressions and various it is quite clear that the choice is only between

types of simultaneous equations. The contents alternatives involving only the recent times.

of the book are akin to much later mathematics, It may be recalled here that there have

mostly of the kind that appeared in school books also been other instances of exposition and

of our times or those of the swamiji’s youth, and propagation of such faster methods of compu-

it is unthinkable, in the absence of any press- tation applicable in various special situations

ing evidence, that they go back to the Vedic (without claims of their coming from ancient

lore. The book really consists of a compila- sources). Trachtenberg’s Speed System (see

tion of tricks in elementary arithmetic and alge- Arther and McShane, 1965) and Lester Meyers’

bra, to be applied in computations with numbers book, High-Speed Mathematics (Meyers, 1947)

and polynomials. By a ”trick” I do not mean a are some well-known examples of this. Tracht-

sleight of hand or something like that; in a gen- enberg had even set up an Institute in Germany

eral sense a trick is a method or procedure which to provide training in high-speed mathematics.

involves observing and exploring some special While the swamiji’s methods are independent of

features of a situation, which generally tend to these, for the most part they are similar in spirit.

be overlooked; for example, the trick described One may wonder why such methods are not

for finding the square of numbers like 15 and 25 commonly adopted for practical purposes. One

with 5 in the unit’s place makes crucial use of main point is that they turn out to be quicker

the fact of 5 being half of 10, the latter being the only for certain special classes of examples. For

base in which the numbers are written. Some of a general example the amount of effort involved

the tricks given in the book are quite interesting (for instance, the count of the individual oper-

and admittedly yield quicker solutions than by ations needed to be performed with digits, in

standard methods (though the comparison made arriving at the final answer) is about the same

in the book are facetious and misleading). They as required by the standard methods; in the

are of the kind that an intelligent hobbyist ex- swamiji’s book, this is often concealed by not

7

tem of mathematics,” and in Chapter XL we come up with. The tricks are, however, based

find a whole list of subjects such as dynam- on well-understood mathematical principles and

ics, statics, hydrostatics, pneumatics and applied there is no mystery about them.

mathematics listed alongside such elementary Of course to produce such a body of tricks,

things as subtractions, ratios, proportions and even using the well-known is still a non-trivial

such money matters as interest and annuities (!), task and there is a serious question of how this

discounts (!) to which we are assured, without came to be accomplished. It is sometimes sug-

going into details, that the Vedic sutras can be gested that Tirthaji himself might have invented

applied. Need we comment any further on this? the tricks. The fact that he had a M.A.degree in

The remaining chapters are mostly elementary mathematics is notable in this context. It is also

in content, on account of which one does not possible that he might have learnt some of the

see such marked incongruities in their respect. tricks from some elders during an early period

It has, however, been pointed out by Shukla that in his life and developed on them during those

many of the topics considered in the book are ”eight years of concentrated contemplation in

alien to the pursuits of ancient Indian mathe- forest solitude:” this would mean that they do in-

maticians, not only form the Vedic period but volve a certain element of tradition, though not

until much later (Shukla, 1989 or Shukla, 1991). to the absurd extent that is claimed. These can,

These include many such topics as factorisation however, be viewed only as possibilities and it

of algebraic expressions, HCF (highest com- would not be easy to settle these details. But

mon factor) of algebraic expressions and various it is quite clear that the choice is only between

types of simultaneous equations. The contents alternatives involving only the recent times.

of the book are akin to much later mathematics, It may be recalled here that there have

mostly of the kind that appeared in school books also been other instances of exposition and

of our times or those of the swamiji’s youth, and propagation of such faster methods of compu-

it is unthinkable, in the absence of any press- tation applicable in various special situations

ing evidence, that they go back to the Vedic (without claims of their coming from ancient

lore. The book really consists of a compila- sources). Trachtenberg’s Speed System (see

tion of tricks in elementary arithmetic and alge- Arther and McShane, 1965) and Lester Meyers’

bra, to be applied in computations with numbers book, High-Speed Mathematics (Meyers, 1947)

and polynomials. By a ”trick” I do not mean a are some well-known examples of this. Tracht-

sleight of hand or something like that; in a gen- enberg had even set up an Institute in Germany

eral sense a trick is a method or procedure which to provide training in high-speed mathematics.

involves observing and exploring some special While the swamiji’s methods are independent of

features of a situation, which generally tend to these, for the most part they are similar in spirit.

be overlooked; for example, the trick described One may wonder why such methods are not

for finding the square of numbers like 15 and 25 commonly adopted for practical purposes. One

with 5 in the unit’s place makes crucial use of main point is that they turn out to be quicker

the fact of 5 being half of 10, the latter being the only for certain special classes of examples. For

base in which the numbers are written. Some of a general example the amount of effort involved

the tricks given in the book are quite interesting (for instance, the count of the individual oper-

and admittedly yield quicker solutions than by ations needed to be performed with digits, in

standard methods (though the comparison made arriving at the final answer) is about the same

in the book are facetious and misleading). They as required by the standard methods; in the

are of the kind that an intelligent hobbyist ex- swamiji’s book, this is often concealed by not

7

8.
writing some of the steps involved, viewing it have had a difficult time with their arithmetic

as ”mental arithmetic.” Using such methods of at school and even those who might have been

fast arithmetic involves the ability or practice to fairly good would have lost touch, the very fact

recognise various patterns which would simplify of someone doing some computations rather fast

the calculations. Without that, one would actu- can make an impressive sight. This effect may

ally spend more time, in first trying to recognise be enhanced with well-chosen examples, where

patterns and then working by rote anyway, since some quicker methods are applicable.

in most cases it is not easy to find useful pat- Even in the case of general examples where

terns. the method employed is not really more efficient

People who in the course of their work have than the standard one, the computations might

to do computations as they arise, rather than appear to be fast, since the demonstrator would

choose the figures suitably as in the demonstra- have a lot more practice than the people in the

tions, would hardly find it convenient to carry audience. An objective assessment of the meth-

them out by employing umpteen different ways ods from the point of view of overall use can

depending on the particular case, as the methods only be made by comparing how many individ-

of fast arithmetic involve. It is more convenient ual calculations are involved in working out var-

to follow the standard method, in which one has ious general examples, on an average, and in

only to follow a set procedure to find the answer, this respect the methods of fast arithmetic do not

even though in some cases this might take more show any marked advantage which would offset

time. Besides, equipment such as calculators the inconvenience indicated earlier. In any case,

and computers have made it unnecessary to tax it would be irrational to let the element of sur-

one’s mind with arithmetical computations. In- prise interfere in judging the issue of origin of ”

cidentally, the suggestion that this ”Vedic math- Vedic mathematics” or create a dreamy and false

ematics” of the Shankaracharya could lead to picture of its providing solutions to all kinds of

improvement in computers is totally fallacious, problems.

since the underlying mathematical principles in- It should also be borne in mind that the book

volved in it were by no means unfamiliar in pro- really deals only with some middle and high

fessional circles. school level mathematics; this is true despite

One of the factors causing people not to what appear to be chapters dealing with some

pay due attention to the obvious questions about notions in calculus and coordinate geometry and

”Vedic mathematics” seems to be that they are the mention of a few, little more advanced top-

overwhelmed by a sense of wonderment by the ics, in the book. The swamiji’s claim that ”there

tricks. The swamiji tells us in the preface how is no part of mathematics, pure or applied, which

”the educationists, the cream of the English edu- is beyond their jurisdiction” is ludicrous. Math-

cated section of the people including highest of- ematics actually means a lot more than arith-

ficials (e.g.the high court judges, the ministers metic of numbers and algebra of polynomials;

etc.) and the general public as such were all in fact multiplying big numbers together, which

highly impressed; nay thrilled, wonder-struck a lot of people take for mathematics, is hardly

and flabbergasted!” at his demonstrations of something a mathematician of today needs to

the ”Vedic mathematics.” Sometimes one comes engage himself in. The mathematics of today

across reports about similar thrilling demonstra- concerns a great variety of objects beyond the

tions by some of the present-day expositors of high school level, involving various kinds of ab-

the subject. Though inevitably they have to be stract objects generalising numbers, shapes, ge-

taken with a pinch of salt, I do not entirely doubt ometries, measures and so on and several com-

the truth of such reports. Since most people binations of such structures, various kinds of

8

as ”mental arithmetic.” Using such methods of at school and even those who might have been

fast arithmetic involves the ability or practice to fairly good would have lost touch, the very fact

recognise various patterns which would simplify of someone doing some computations rather fast

the calculations. Without that, one would actu- can make an impressive sight. This effect may

ally spend more time, in first trying to recognise be enhanced with well-chosen examples, where

patterns and then working by rote anyway, since some quicker methods are applicable.

in most cases it is not easy to find useful pat- Even in the case of general examples where

terns. the method employed is not really more efficient

People who in the course of their work have than the standard one, the computations might

to do computations as they arise, rather than appear to be fast, since the demonstrator would

choose the figures suitably as in the demonstra- have a lot more practice than the people in the

tions, would hardly find it convenient to carry audience. An objective assessment of the meth-

them out by employing umpteen different ways ods from the point of view of overall use can

depending on the particular case, as the methods only be made by comparing how many individ-

of fast arithmetic involve. It is more convenient ual calculations are involved in working out var-

to follow the standard method, in which one has ious general examples, on an average, and in

only to follow a set procedure to find the answer, this respect the methods of fast arithmetic do not

even though in some cases this might take more show any marked advantage which would offset

time. Besides, equipment such as calculators the inconvenience indicated earlier. In any case,

and computers have made it unnecessary to tax it would be irrational to let the element of sur-

one’s mind with arithmetical computations. In- prise interfere in judging the issue of origin of ”

cidentally, the suggestion that this ”Vedic math- Vedic mathematics” or create a dreamy and false

ematics” of the Shankaracharya could lead to picture of its providing solutions to all kinds of

improvement in computers is totally fallacious, problems.

since the underlying mathematical principles in- It should also be borne in mind that the book

volved in it were by no means unfamiliar in pro- really deals only with some middle and high

fessional circles. school level mathematics; this is true despite

One of the factors causing people not to what appear to be chapters dealing with some

pay due attention to the obvious questions about notions in calculus and coordinate geometry and

”Vedic mathematics” seems to be that they are the mention of a few, little more advanced top-

overwhelmed by a sense of wonderment by the ics, in the book. The swamiji’s claim that ”there

tricks. The swamiji tells us in the preface how is no part of mathematics, pure or applied, which

”the educationists, the cream of the English edu- is beyond their jurisdiction” is ludicrous. Math-

cated section of the people including highest of- ematics actually means a lot more than arith-

ficials (e.g.the high court judges, the ministers metic of numbers and algebra of polynomials;

etc.) and the general public as such were all in fact multiplying big numbers together, which

highly impressed; nay thrilled, wonder-struck a lot of people take for mathematics, is hardly

and flabbergasted!” at his demonstrations of something a mathematician of today needs to

the ”Vedic mathematics.” Sometimes one comes engage himself in. The mathematics of today

across reports about similar thrilling demonstra- concerns a great variety of objects beyond the

tions by some of the present-day expositors of high school level, involving various kinds of ab-

the subject. Though inevitably they have to be stract objects generalising numbers, shapes, ge-

taken with a pinch of salt, I do not entirely doubt ometries, measures and so on and several com-

the truth of such reports. Since most people binations of such structures, various kinds of

8

9.
operations, often involving infinitely many en- It is often claimed that ” Vedic mathemat-

tities; this is not the case only about the fron- ics” is well-appreciated in other countries, and

tiers of mathematics but a whole lot of it, includ- even taught in some schools in UK etc.. In the

ing many topics applied in physics, engineering, normal course one would not have the means to

medicine, finance and various other subjects. examine such claims, especially since few de-

Despite all its pretentious verbiage page af- tails are generally supplied while making the

ter page, the swamiji’s book offers nothing claims. Thanks to certain special circumstances

worthwhile in advanced mathematics whether I came to know a few things about the St. James

concretely or by way of insight. Modern Independent School, London which I had seen

mathematics with its multitude of disciplines quoted in this context. The School is run by the

(group theory, topology, algebraic geometry, ’School of Economic Science’ which is, accord-

harmonic analysis, ergodic theory, combinato- ing to a letter to me from Mr. James Glover, the

rial mathematics-to name just a few) would be a Head of Mathematics at the School, ”engaged in

long way from the level of the swamiji’s book. the practical study of Advaita philosophy”. The

There are occasionally reports of some ”re- people who run it have had substantial involve-

searchers” applying the swamiji’s ”Vedic math- ment with religious groups in India over a long

ematics” to advanced problems such as Kepler’s period. Thus in essence their adopting ” Vedic

problem, but such work involves nothing more mathematics” is much like a school in India run

than tinkering superficially with the topic, in the by a religious group adopting it; that school be-

manner of the swamiji’s treatment of calculus, ing in London is beside the point. (It may be

and offers nothing of interest to professionals in noted here that while privately run schools in In-

the area. dia have limited freedom in choosing their cur-

Even at the school level ”Vedic mathemat- ricula, it is not the case in England). It would be

ics” deals only with a small part and, more im- interesting to look into the background and mo-

portantly, there too it concerns itself with only tivation of other institutions about which similar

one particular aspect, that of faster computation. claims are made. At any rate, adoption by in-

One of the main aims of mathematics education stitutions abroad is another propaganda feature,

even at the elementary level consists of devel- like being from ancient source, and should not

oping familiarity with a variety of concepts and sway us.

their significance. Not only does the approach It is not the contention here that the con-

of ” Vedic mathematics” not contribute anything tents of the book are not of any value. Indeed,

towards this crucial objective, but in fact might some of the observations could be used in teach-

work to its detriment, because of the undue em- ing in schools. They are entertaining and could

phasis laid on faster computation. The swamiji’s to some extent enable children to enjoy math-

assertion ”8 months (or 12 months) at an aver- ematics. It would, however, be more appro-

age rate of 2 or 3 hours per day should suffice priate to use them as aids in teaching the re-

for completing the whole course of mathemati- lated concepts, rather than like a series of tricks

cal studies on these Vedic lines instead of 15 or of magic. Ultimately, it is the understanding

20 years required according to the existing sys- that is more important than the transient excite-

tems of the Indian and also foreign universities,” ment, By and large, however, such pedagogical

is patently absurd and hopefully nobody takes it application has limited scope and needs to be

seriously, even among the activists in the area. made with adequate caution, without being car-

It would work as a cruel joke if some people ried away by motivated propaganda.

choose to make such a substitution in respect of It is shocking to see the extent to which

their children. vested interests and persons driven by mis-

9

tities; this is not the case only about the fron- ics” is well-appreciated in other countries, and

tiers of mathematics but a whole lot of it, includ- even taught in some schools in UK etc.. In the

ing many topics applied in physics, engineering, normal course one would not have the means to

medicine, finance and various other subjects. examine such claims, especially since few de-

Despite all its pretentious verbiage page af- tails are generally supplied while making the

ter page, the swamiji’s book offers nothing claims. Thanks to certain special circumstances

worthwhile in advanced mathematics whether I came to know a few things about the St. James

concretely or by way of insight. Modern Independent School, London which I had seen

mathematics with its multitude of disciplines quoted in this context. The School is run by the

(group theory, topology, algebraic geometry, ’School of Economic Science’ which is, accord-

harmonic analysis, ergodic theory, combinato- ing to a letter to me from Mr. James Glover, the

rial mathematics-to name just a few) would be a Head of Mathematics at the School, ”engaged in

long way from the level of the swamiji’s book. the practical study of Advaita philosophy”. The

There are occasionally reports of some ”re- people who run it have had substantial involve-

searchers” applying the swamiji’s ”Vedic math- ment with religious groups in India over a long

ematics” to advanced problems such as Kepler’s period. Thus in essence their adopting ” Vedic

problem, but such work involves nothing more mathematics” is much like a school in India run

than tinkering superficially with the topic, in the by a religious group adopting it; that school be-

manner of the swamiji’s treatment of calculus, ing in London is beside the point. (It may be

and offers nothing of interest to professionals in noted here that while privately run schools in In-

the area. dia have limited freedom in choosing their cur-

Even at the school level ”Vedic mathemat- ricula, it is not the case in England). It would be

ics” deals only with a small part and, more im- interesting to look into the background and mo-

portantly, there too it concerns itself with only tivation of other institutions about which similar

one particular aspect, that of faster computation. claims are made. At any rate, adoption by in-

One of the main aims of mathematics education stitutions abroad is another propaganda feature,

even at the elementary level consists of devel- like being from ancient source, and should not

oping familiarity with a variety of concepts and sway us.

their significance. Not only does the approach It is not the contention here that the con-

of ” Vedic mathematics” not contribute anything tents of the book are not of any value. Indeed,

towards this crucial objective, but in fact might some of the observations could be used in teach-

work to its detriment, because of the undue em- ing in schools. They are entertaining and could

phasis laid on faster computation. The swamiji’s to some extent enable children to enjoy math-

assertion ”8 months (or 12 months) at an aver- ematics. It would, however, be more appro-

age rate of 2 or 3 hours per day should suffice priate to use them as aids in teaching the re-

for completing the whole course of mathemati- lated concepts, rather than like a series of tricks

cal studies on these Vedic lines instead of 15 or of magic. Ultimately, it is the understanding

20 years required according to the existing sys- that is more important than the transient excite-

tems of the Indian and also foreign universities,” ment, By and large, however, such pedagogical

is patently absurd and hopefully nobody takes it application has limited scope and needs to be

seriously, even among the activists in the area. made with adequate caution, without being car-

It would work as a cruel joke if some people ried away by motivated propaganda.

choose to make such a substitution in respect of It is shocking to see the extent to which

their children. vested interests and persons driven by mis-

9

10.
guided notions are able to exploit the urge for educate people on the truth of this so-called

cultural self-assertion felt by the Indian psy- Vedic mathematics and prevent the use of pub-

che. One would hardly have imagined that a lic money and energy on its propagation, beyond

book which is transparently not from any an- the limited extent that may be deserved, lest the

cient source or of any great mathematical signif- intellectual and educational life in the country

icance would one day be passed off as a store- should get vitiated further and result in wrong

house of some ancient mathematical treasure. attitudes to both history and mathematics, espe-

It is high time saner elements joined hands to cially in the coming generation.

[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 Mathemat-

ica, 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 Educa-

tion, Vol 5: No. 3, January-March 1989.

[8] K.S. Shukla, Mathematics — The Deceptive Title of Swamiji’s Book, in Issues in Vedic Math-

ematics, (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.

10

cultural self-assertion felt by the Indian psy- Vedic mathematics and prevent the use of pub-

che. One would hardly have imagined that a lic money and energy on its propagation, beyond

book which is transparently not from any an- the limited extent that may be deserved, lest the

cient source or of any great mathematical signif- intellectual and educational life in the country

icance would one day be passed off as a store- should get vitiated further and result in wrong

house of some ancient mathematical treasure. attitudes to both history and mathematics, espe-

It is high time saner elements joined hands to cially in the coming generation.

[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 Mathemat-

ica, 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 Educa-

tion, Vol 5: No. 3, January-March 1989.

[8] K.S. Shukla, Mathematics — The Deceptive Title of Swamiji’s Book, in Issues in Vedic Math-

ematics, (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.

10