This article appeared in the Indian Journal ‘Sambodhi’ Vol. XXIII, 2000 and is reproduced here with permission.

Diacritical marks have not been copied, nor have the few Sanskrit/Hindu lines of text. This article has been scanned into text and though we have done our best there may still be a few errors. For the full text please see the Journal version.

VEDIC SOURCES OF THE 'VEDIC MATHEMATICS'

Dr. N. M. Kansara
Director, Akshardham Centre for Applied Research in Social Harmony (AARSH), Akshardham, Gandhinagar - (382 020)

Jagadguru Shankaracharya Swami Shri Bharati Krishna Tirthaji Maharaja of Govardhan Peeth Matha, Puri, wrote or dictated a book entitled `Vedic Mathematics' based on 29 Sutras, of which 16 deal with the ‘general case’, while the rest 13 treat the special cases.

Looking to the Sutras themselves, they seem to differ from similar ones in other Sutra works in the point that they do not seem to constitute a single compact text of some such work, because they do not contain any reference as to the subject or any statement about the purpose, terminology, extent of the work and etc. They seem to be rather stray Sutras collected from a body of a text, and since they are divided by the Swamiji into two sets, viz., Principal ones and the subordinate ones, our conjecture as stated above is rather supported, because in the Sutra works the system is to start the work with the words `Atha' and end it with the repetition of the last word of the last sutra, with the concluding word `Iti'. Thus, it seems the Swamiji culled his Sutras from some Sulba work, and discovered his own mathematical significance and interpretation of them all, and presented his discovery in the form of the Sutras and their application to various mathematical problems, along with the `proofs' as expected from a veteran mathematician like him.

The declaration of the Sutras as "Vedic" or as belonging to the Vedas, particularly to the Atharvaveda, and his claim that "the Sutras (aphorisms) apply to and cover each and every part of each and every chapter of each and every branch of mathematics including arithmetic, algebra, geometry ­plane and solid, trigonometry - plane and spherical, conics - geometrical and analytical, astronomy, calculus - differential and integral etc., etc.", and that "there is no part of mathematics, pure or applied, which is beyond their jurisdiction" has raised a controversy amongst the mathematicians of India, some of whom have questioned the Vedicity of the Sutras on the ground of their language, and the level of mathematics it deals with. It is endeavoured here to deal with the problem in all possible aspects, and examine the validity or otherwise of the claim.

In order to deal with this problem from all possible viewpoints, and in all possible aspects, we have approached the problem with the following objectives, viz., (i) to try to trace the handwritten manuscript (note-books) of Swami Shri Bharati Krishna Tirthaji Maharaja comprising sixteen volumes of the work, which is supposed to have devoted one volume each for each of the sixteen principal Sutras of `Vedic Mathematics', as given by him; (ii) to try to trace the Vedic Mathematics Sutras from the extent Vedic texts; (iii) to trace the terminology of the Vedic Mathematics Sutras from the Vedic texts; (iv) to examine the validity of the term `Vedic' as applied to the Vedic Mathematics, Sutras; (v) to collate the mathematical data from the Vedic literature; (vi) to compare the VM terminology with that of the medieval mathematicians; (vii) and to compare the mathematical functions of the Sutras with those of the medieval mathematicians.

Findings :

I. About the Handwritten Manuscript Notebooks of BKTM on VM in sixteen volumes:

In his "Author's Preface" to the "Vedic Mathematics or Sixteen Simple Mathematical Formulae from the Vedas (For One-line Answers to All Mathematical Prohlems"¹, H. H. Shri Jagad-Guru Shankaracharya (the late) Swamy Shri Bharti Krishna Tirthaji Maharaja of the Govardhana Peetha Mutt, Puri, has declared, himself about "Vedic Sutras dealt with in the 16 volumes" on Vedic Mathematics.² Smt. Manjulaben Trivedi, the Hon. General Secretary of Sri Vishva Punarnirmana Sangha, Nagpur, who has been a very close and favourite disciple along with her late husband Shri Chimanbhai M. Trivedi, of Shri Jagadguruji BKTM, has noted in the introductory article to the VM³ that the revered Guruji used to say that he had reconstructed the sixteen mathematical formula (given in the text) from the Atharvaveda after arduous research, and `Tapas' of about eight years in the forest of Sringeri.

From the life-sketch given by Smt. Manjulaben,⁴ it seems that, as Venkatraman, BKTM was born in March 1884, passed his M.A. Examination of the American College of Science, Rochester, New York, with Sanskrit, Philosophy, English, Mathematics, History and Science, securing the highest honours in all the seven subjects. As Professor Venkatraman Saraswati, he started his public life under the guidance of Late Hon'ble Shri Gopal Krishna Gokhale, C.I.E. in 1905 in connection with the National Education Movement and the Sourth African issue. Due to his deepest attraction towards the study and practice of the science of sciences - the holy ancient Indian spiritual science or Adhyatma-Vidya, he proceeded, in 1908, to Sringeri Math in Mysore to lay himself at the feet of the renowned late Jagadguru Shankaracharya Maharaj Shri Satchidananda Shivabhinava Nrisimha Bharati Swami. In 1908 or so he had to assume the post of the first Principal of the newly started National College at Rajmahendri under a pressing and clamant call of duty from the nationalist leaders. But after three years, in 1911, he went back to Shri Satchidananda Shivabhainava Nrisimha Bharati Swami at Sringeri.

The next eight years he spent in the profoundest study of the most advanced Vedanta Philosophy and practice of the Brahma-sadhana. In 1919 he was initiated into the holy order of Samnyasa at Varanasi by H. H. Jagadguru Shankaracharya Shri Trivikrama Tirthaji Maharaja of Sharadapeetha and was given the new name, Swami Bharati Krishna Tirtha. And in 1921, he was installed on the pontifical throne of Sharda Peetha Shankaracharya, and in 1925 he shifted to Puri when he was installed as Jagadguru Shankaracharya of the Govardhan Math, while Shri Swarupanandaji was installed on the Shardapeetha Gadi. In 1953, he founded at Nagpur an institution named Shri Vishwa Punarnirmana Sangha (World Reconstruction Association), with Shri Chimanlal Trivedi as the General Secretary and the Administrative Board consisted of his disciples, devotees and admirers.⁵

Thus, it seems he discovered the VM Sutras during his stay at Shringeri, between 1911 and 1919, and at the age of his 34th or 35th year and for next few years he was busy working on these Sutras, and he seems to have definitely written, in school notebooks, all of his sixteenth volumes treating each of his sixteen Sutras in one independent volume, most probably well before 1953.

From Shri P. M. Trivedi,⁶ who is related to both Shri Chimanlal Trivedi and Smt. Manjulaben Trivedi, informed me that originally he himself and the latter two belonged to Kapadwanj (District Kheda, Gujarat), and perhaps they turned into devotees of BKTM, when as the Jagadguru Shankaracharya he visited Kapadwanj for the monsoon months of the year 1937 or so. During his stay at Kapadwanj, BKTM is said to have borrowed some amount from Shri Manilal Desai, a money-lender of Dakor and devotee of him, and as a security against the borrowed amount, he had pledged his manuscript note­books safely stuffed and packed in a few tin trunk boxes. As a security containing the life-long research work of his highly revered Shankaracharya, the boxes were scrupulously guarded and preserved by Shri Manibhai who had spread his bed on the place under or beneath which the boxes were kept. It seems, BKTM somehow could not repay the amount to Shri Manilal Desai till the end of the latter's lifetime, say till about the year 1955 or so. And then, the boxes came under the charge of Shri Laxminarayan, the son of Shri Manilal Desai. Shri Laxminarayan got them transported from Dakor to his residence at Asarva in Ahmedabad.

Professor Vijaya M. Sane,⁷ who evinced a great interest in tracing the location of the boxes containing the manuscript note-books of the VM in sixteen volumes, took great pains in utilising his contacts with the higher­ up in the Gujarat Government and Gujarat Police Department, and finally located the residence of Shri Laxminarayan in Ahmedabad, got in touch with the then Jagadguru Shankaracharya Swami Shri Abhinava Sachidananda Tirthaji Maharaj of Sharada Peetha, Dwarka (Gujarat), got the boxes confiscated and searched. But the boxes were found to contain useless scraps and old shoes and such other trash. After that when Prof. Sane contacted Shri Laxminarayan personally and talked to him, he was told that some German scholar had came searching for him and had offered big amount for the VM manuscript material of the sixteen volumes; that he had sold the contents of the boxes to that German scholar for Rs. 80,000/-, and had stuffed the boxes with rubbish. Even after that Shri Sane tried his utmost to trace the whereabouts and identity of the German scholar, through the help of Hon'ble Shri Hitendrabhai Desai, the then Chief Minister of Gujarat, but his efforts donot seem to have met with any degree of success so far. This must have happened some time in the year 1955-56, since, "unfortunately, the said manuscripts were lost irretrievably from the place of their deposit and this colossal loss was finally confirmed in 1956".⁸

And as has been declared by Smt. Manjulaben Trivedi,⁹ her husband Shri Chimanlal Trivedi, found that Secretary of the Sri Vishva Punarnirmana Sangha, found that as a result of demonstrations before learned people and societies by BKTM about the Vedic Mathematics as discovered by him, some people who had grasped a smattering of the new Sutras had already started to dazzle audiences as prodigies claiming occult powers without acknowledging indebtedness to the Sutras of Jagadguruji, and he pleased earnestly with Gurudeva and persuaded him to arrange for the publication of the Sutras in his own name. It was finally in 1957, when he decided to undertake a tour of the U.S.A., that he re-wrote from memory the present volume, viz., the VM (1965 Edn.), giving an introductory account of the sixteen formulae reconstructed by him; he wrote down the volume in his old age within one month and a half with his failing health and weak eyesight. The type-script of the VM was left over by BKTM in U.S.A. in 1958 for publication. It was through the good offices of Justice. N. H. Bhagavati, the then Vice Chancellor of the Banaras Hindu University, and Dr. Pandit Omkarnath Thakur, the veteran classical musician, that the Banaras Hindu University published the VM in the Nepal Endowment Hindu Vishva­vidyalaya Sanskrit Granthamala (Vol. 10), in 1965, after about five years since the demise of BKTM in 1960.

Thus, it seems to be a solid fact that BKTM did write down his sixteen volumes on the sixteen Sutras of the Vedic Mathematics, that he deposited his manuscript with his devotee Shri Manilal Desai of Dakor in Gujarat, that after the death of Shri Manilal, the material came in possession of his son Laxminarayan Desai, and the latter sold it to some German scholar for Rs. 80,000/-, and that, at least subconsciously, BKTM was under the impression that he was writing the book overall again, in a series of a number of volumes, although what he could write was a single volume published as the VM.¹⁰

On the Vedic Sources of the VM Sutras:

In his talk and demonstration given to a small group of student mathematicians at the California Institute of Technology, Pasedena, California on l9th February, 1958, BKTM has been recorded to have said "that I also speak summarily about mathematics which I have been able to get from the Sutras of the Atharvaveda."¹¹ Giving some further details, he said: "one particular portion I am referring to, a particular portion of the Atharva-Veda is called the ganita sutras. The ganita sutras are also called the Sulba Sutras `the easy mathematical formulae', that's the meaning of the expression. And there are sixteen sutras, sixteen aphorisms in all, and the general name ‘ganita' mathematics is given to the subject."¹² In the same talk he said: "And, then fourthly, in the Atharvaveda we have what is called the Sthapatyaveda which is the Sanskrit term for a combination of sciences starting with mathematics and all its branches without a single exception going on to the application of mathematics in various other departments, including architecture, engineering, and so forth."¹³ According to him, there are four Upavedas, viz., Ayurveda, Dhanurveda, Gandharvaveda and Sthd­patyaveda, connected with the Rigveda, Yajurveda, Samaveda and Atharva­veda, respectively.¹⁴

Under the sub-topic "Ganita Sutras",¹⁵ BKTM has given the following details : "One particular portion of the Atharvaveda is called the Ganita­sutras. They are also called the Sulba-sutras, and there are sixteen, aphorisms in all. In this connection, he referred to Professor Colebrooke, and quoted him as having said to the effect that he was, unable to understand what the contents of those sutras are, and what connection those sutras have with mathematics, and that he did not understand those sutras; that it was unintelligible to him, it was beyond him: He has further informed us that coming to the same passage, the same portion of the Atharvaveda, Horace Hayman Wilson remarked "this is all nonsense", and that R. T. Griffith said it was "utter nonsense".¹⁶

We are further informed by BKTM that the above remarks of Colebrooke, Wilson and Griffith put him on the track, since, he thought that there , must be something in the subject which was being discussed with so much earnestness and which the commentators were trying to understand but could make nothing out of. So he went on with his simple idea that there was some meaning The meaning may be all absolutely wrong; but to dismiss something off hand as nonsense because it is not understandable, was not correct.¹⁷

Sanskrit has a certain peculiarity about it, that the same passage very often deals with a different subject and is capable of yielding different. meanings relating to different subjects. For example, there is a hymn of praise addressed to Sri Krsna.¹⁸ This verse also gives the value of p/10 to thirty ­two decimal places. The literal meaning of the verse is that in the reign of King Kamsa unsanitary conditions prevailed, which has apparently nothing to do with mathematics.

He further clarified that `Mathematical formulae' is the heading of the subject, and inside we are told that the tyrant king ruled over the people oppressively.<sup>19</sup> And here too, the heading is `Ganita Sutras', mathematics formulae. So he thought there must be something. And for long years of meditation in the forest, and intense study of the lexicographies, lexicons of earlier times,²⁰ he devoted himself to the task of discovering the mathematical meaning of the ganita-sutras. He studied the old lexicons, including Visva, Amara, Arnava, Sabdakalpadruma, etc. With these he was able to find out the meanings; he got the key in that way in one instance, and one thing after another helped him in elucidation of the other sutras, the other formulae. And he found to his extreme astonishment and gratification that the sutras dealt with mathematics in all its branches; that only sixteen sutras cover all the branches of mathematics, arithmetic, algebra, geometry, trigonometry, physics, plain and spherical geometry, conics, calculus, both differential and integral, applied mathematics of various kinds dynamics, hydrostatics, statics, kinematics, and all.²¹

In our endeavour to locate the portions of the Atharvaveda, that confronted Colebrooke, Wilson and Griffith and evoked the above-mentioned remarks from them, we have scanned through all the writings of H. Th. Colebrooke, which being more than a century old, could be available only in the very old libraries at Pune and Bombay, for personal reference and verification.

As regards Colebrooke, he is known to have remarked as follows:²² ".. the Vedas ... are too voluminous for a complete translation of the whole; and what they contain would hardly reward the labour of the reader; much less that of a translator. The ancient dialect in which they are composed, and especially that of the first three Vedas, is extremely difficult and obscure; ... its difficulties must long continue to prevent such an examination of the whole Vedas, as would be requisite for extracting all that is remarkable and important in those voluminous works." During the course of his essays he has quoted generally from the Atharvaveda Samhita of the Saunakiya Sakha.

Horace Hayman Wilson, too, has referred to Colebrooke's opinion in his essays and lectures.²³ The portion of the Atharvaveda generally referred to by him seems to be the last two books, i.e. the Kandas XIX and XX.

R. T. Griffith, who has translated the whole of the Atharvaveda in English²⁴ refers to "some attempts to bring sense out of utter nonsense which constitutes part of the last two books" of the Atharvaveda.²⁵

Now, looking into the Griffith's translation and notes of the last two books of the Atharvaveda, we find him passing the following remarks, in the indicated places:

AV. XIX, 6, Vol. II, p. 265, f.n. - "This hymn is generally called the Purusha Sukta or Purusa hymn, ... The Rgvedic hymn has been translated also by Colebrooke, Miscellaneous Essays, pp. 167-168 ... Wilson's Translation should be consulted for the views of Sayana and the Indian scholars of his own and earlier times.

AV. XIX, 22, 6, Vol. II, p. 280, f.n. - "6 Small ones the Ksudras. Various portions and hymns of the Atharvaveda, which are not clearly identifiable, are designated by these and the remaining fantastic names."

AV. XX, 5, 6, Vol. II, p. 323, f.n. - "6 Famed for thy radiance, worshipped well . the words this rendered, Sacigo and Sacipgjana, have not been satisfactorily explained by the commentator, and their meaning is still uncertain."

AV. XX, 5, 7, Vol. II, p. 324, f. n. 7 continued - "See Professor Wilson's note who observes that the construction is loose, and the explanation not very satisfactory."

AV. XX, 16, 3, Vol. II, p. 332, f. n. 3 - "Prof. Wilson renders Sthivibhyah in this place by `from the granacies'."

AV. XX, 16, 9, Vol. II, p. 333, f. n. 9 - "Prof. Wilson following Sayana, paraphrases the second line:-“.

AV. XX, 34, 12, Vol. p. 351, f. n. - "12 The stanza is not taken from the Rgveda; and the manuscripts on which the printed text is based are corrupt and unintelligible as they stand."

AV. XX, 35, 7, Vol. II, p. 353, f. n. - "7 The verse is difficult. Sayana, Wilson, Benfey and Grassmann take Visnu to be an appellative or epithet of Indra."

AV. XX, 58, 1, Vol. II, p. 374 f.n. 3 - ."1 This stanza is difficult and obscure ... See Prof. Cowell's note in Wilson's Translation."

AV. XX, 74, 3, Vol. II, p. 389, f. n. - "The text is very elliptical and obscure. ... ... - Wilson."

AV. XX, 76, 1, Vol. II, p. 391, f. n. - "1 The meaning of the stanza is obscure; and the text of the first half line is unintelligible. I follow the reading which Sayana gives in his Commentary, Vayo instead of Va yo ... - Wilson."

AV. XX. 133, Vol. II, p. 441, f. n. - "There are five more stanzas, all with the refrain : `Maiden, it truly is not so as thou, O maiden, fanciest.' A mere literal translation of these would be unintelligible, and the matter does not deserve expansion or explanation. These six stanzas are called Pravahlikas or Enigmatical Verses."

Thus, BKTM seems to have referred to some portion of the Books XIX and XX of the Atharvaveda, while he passed his above comments.

Now, when we search for the VM sutras of BKTM in both the Saunakiya and the Paippalada Samhitas, we find the passages like the following :

AV. XIX, 6, 1-3:

[Six lines of Sanskrit quoted here]

AV. XIX, 8, 2:

[Line of Sanskrit quoted here]

AV. XIX, 22:

[Two lines of Sanskrit quoted here]

AV. XIX, 23:

[Three lines of Sanskrit quoted here]

AV. XIX, 32, 1 .

[Line of Sanskrit quoted here]

AV. XIX, 33, l .

[Line of Sanskrit quoted here]

But nowhere in the Atharvaveda Samhita, of both the Saunakiya or the Paippalada Saakhas, nor in Sayana's Commentary, nor in the Gopatha Brahmana, nor in the Kausika Sutra, nor in the Vaitana Sutra, nor in any of the Atharvaveda-parisistas, nor in the Angiras-kalpa, nor in the Karma­panjika, Karma-samuccaya, nor in the Atharvana-rahasya do we find any of the sutras as they are given by BKTM.

In this connection, BKTM has referred to Sthapatyaveda which is said to be an Upaveda of the Atharvaveda. Monier-Williams²⁶ notices the following under the word `Upaveda':

"upa-veda, as, m. `secondary knowledge', N. of a class of writings subordinate or appended to the four Vedas (viz., the Ayurveda or science of medicine, to the Rgveda; the Dhanurveda or science of archery, to the Yajurveda; the Gandharvaveda or science of music, to the Samaveda; and the Sastraveda or science of arms, to the Atharvaveda; this is according to the Caranavyuha, but Susruta and Bhavaprakasa make the Ayur-veda belong to the Atharvaveda; according to others, the Sthapatyaveda or science of architecture, and Silpasastra or knowledge of arts, are reckoned as the fourth Upaveda."

The portion of the Atharvaveda, called Ganita-Sutra, according to BKTM, is also known as `Sulba-sutra' and it is said to belong to the Atharvaveda, and deal with the art and science of the building of fire-altars. Hence it would rightly be a part of the Sthapatyaveda.

We should note here that the religious rites of the Paippaladins, dealt with in their Kalpa and Paddhati - the Angirasa-kalpa and the Karma-panjika ­that have come down to us, are all of a Grhya character. They are to be performed on a single fire (ekagni-sadhya), and not three, as is necessary for the Srauta rites. There was a Paippalada Srauta Stltra in seven chapters, written by Agastya, as is known from a statement to this effect appearing in the Prapanca-hrdaya.²⁷ Like the Kalpasutras of Asvalayana, Bodhayana and Jaimini, the work of Agastya also contained treatment of distinctive Vedic rituals. The lost Srauta Sutra of the Paippalada Sakha may, however, be taken to have contained more sacrificial matters than what the Vaitana Sutra of the Saunaka Sakha does. The Paippalada Samhita itself has a large number of sacrificial hymns which are not found in the Saunaka Sarhhita. These hymns relate to the Srauta rites like the Darsa-paurnamasa, Agnyadheya and Gostoma.²⁸ Perhaps, the portion of the Atharvaveda known to BKTM as `Ganita-sutra' might belong to the Kalpasittra part of this Srautasutra of Agastya. But, unfortunately, we have yet to come across a Sulba Sutra attached to the Atharvaveda; none else, except BKTM, has as yet noticed any such work. The question of the location and the veracity of the `Ganita-sutra' being a part of some Kalpasiitra of the Atharvaveda, should remain open till we discover it with the help of some of the oral reciters of the Paippalada Atharvaveda.

The culture of the Paippalada Atharvaveda considered so long associated with Kashmir, but totally extinct from our country at the present time, is still a living force in the Eastern region of India. Thousands of Paippaladins residing in Orissa and the adjacent parts of Bihar and West Bengal have survived to this day unnoticed by the scholarly world. They have kept themselves away from the public eye and so much so that even the Sanskrit Commission appointed by the Government of India in 1956, unaware of their existence in the state of Orissa, was unable of their existence in the state of Orissa, was unable to locate them during its tours in the State. These people still follow their traditional rites and customs. Valuable Paippalada works which could not, up to this time, be traced anywhere in the country, have been found carefully preserved in their safe custody.²⁹

Now, as regards the Sulba Sutras. They are the manuals for the construction of altars which are necessary in connection with the sacrifices of the Vedic Hindus. The Sulba Sutras are sections of the Kalpasutras, more particularly of the Srautasutras, which are considered to form one of the six Vedangas or "The Members of the Veda", and deal specially with rituals or ceremonials. Each Srautasutra seems to have its own Sulba section. So there were, very likely, several such works in ancient times.³⁰ Patanjali, the Great commentator of Panini’s Grammar, states that there were as many as 1131 or 1137 different schools of the Veda; in particular, there were 21 different schools of the Rgveda, 101 schools of the Yajurveda, 1,000 of the Samaveda, and 9 Or 15 of the Atharvaveda.³¹ But most of them are now lost.

At present we know, however, of only seven Sulba-stitras, those belonging to the Srauta-sutras of Baudhayana, Apastamba, Katyayana, Manava, Maitrayana, Varaha and Vadhula, while we find only the reference to two other works, viz., Masaka and Hiranyakesi, in the commentary of Karvindaswami on the Apastamba Sulba (xi. 11). As related to the different Vedas, the Sulba-sutras of Baudhayana, Apastamba, Manava, Maitrayana and Varaha belong to the Krsna Yajurveda; and the Katyayana Sulba-sutra to the Sukla Yajurveda.³²

As regards their importance, the available Sulba-sutras can sharply be divided into two classes. The first class will include the manuals of Baudhayana, Apastamba md Katyayana. They give us an insight into the early state of Hindu geometry before the rise and advent of the Jaina sect (500-300 B.C.). The Sulba-sutras of Manava, Varaha, Maitrayana and Vadhula, comprising the second class, add particularly very little to our stock of information in this respect.³³

It was perhaps primarily in connection with the construction of the sacrificial altars of proper size and shapes that the problems of geometry and also of arithmetic and algebra presented themselves, and were studied in ancient India, just as the study of astronomy is known to have begun and developed out of the necessity for fixing the proper time for the sacrifices and agriculture. At any rate, from the Sulba-sutras we get a glimpse of the knowledge of geometry that the Vedic Hindus had. Incidentally they furnish us with a few other subjects of much mathematical interest.³⁴

Vedic Sources of the VM-Sutra Terminology :

The VM sutras contain some very common mathematical terminology, which has so far been hardly examined from this point of view. Dr. Satyakama Varma has concluded in his research paper³⁵ that though the term employed in these sutras cannot be claimed to be of the Vedic origin, yet they are later synonyms of the equivalent original Vedic terms, that the Vedic texts include much of the scientific and mathematical statements, which can make a strong basis for such like sutras, and that when the Jagad Guru claims that he has adopted nothing but Vedic Mathematics, he is right in his own way. On closer examination, we find that most of the terms utilised in the VM sutras are found to be prevalent in the Srauta Sutras and the Sulba Sutras. We shall see here which terms are used in which Vedic texts :

(1) ANTA - Bau. Sul. I. 23.

(2) ANTYA - Tait. Sam. 1. 7. 9. 1.; Maitr. Sam. 1.11.3; "the last member of a mathematical series" - Monier-Williams, Ske.-Eng. Dict., p. 44, Col. 3 : Ap. Sul. 2.3; Ath. Anu. 20.9; 104; 118.

(3) ADYA - Ath. Sam. 19.22.1.

(4) ANURtiPYA / ANURtJPATVA - Ap. Sul. 13.8.

(5) URDHVA / URDHVAPRAMANA - Ap. Sul. 9.15; Bau. Sul. 2.3; Vadh. Sul. 7.2; 11. 9.; Hir. Sul. 3.14; 4.20.

(6) LTNIKRTYA / LlNIKAROTI - Nid. Su. 1.7.28.

(7) EKANYilNA / EKONA - Kat. Sul. 6.7.

(8) EKADHIKA - Kat. Sul. 6.7.

(9) GUNAKA / GUNA / DVIGUNA - Bau. Sul. 1.30.

(10) GUMTA - Madh. Si. 16.2.; Yajn. Si. 2.104-105. .

(11) CARAMA - Ksud. Su: 1.1.; Bau. Sr. 10.48.2.

(12) TIRYAK / TIRYAG-DVIGUNA - Kat. Sul. 6.7; TIRYAG-BHEDA Bau. Sul. 17.8; 19.7; TIRYAN-MANA Kat. Sul. 7.32; Bau. Sul. 1.46.

(13) DASAKA - Ath. Anu. 19.17.

(14) DASATAH - Asv. Sr. 8.5.7; Ap. Sr. 20.14.2; 22.17.5; Hir. Sr. 17.6.42. (15) NAVATAH - Asv. Sr. 8.5.7; Bau. Sr. 15.23.

(16) NIKHILA - Madhy. Sr. 1.7.8.10.

(17) NYLlNA - Asv. Sr. 1.11.15.

(18) PURANA - Kat. Sr. 24.7.18; Ap. Sr. 16.26.9; 16.27.6; PURAYET Bau. Sul. 8.12

(19) MADHYAMA - Sankh. Sr. 7.27.20; MADHYAMA-PURVA Bau. Sul. 7.17. (20) YAVAT... TAVAT - YAVAT-PRAMANA - Kat. Sul. 37.12; Ap. Sul. 3.11; Hir. Sul. 1.50; TAVAD-ANTARALA Bau. Sul. 8.11; Hir. Sul. 3.17.

(21) YOJAYET / YUJYATE - Hir. Sul. 4.20.; Ap. Sul. 11.46; Bau. Sul. 21.7.

(22) LOPANA / LOPA - Kat. Sr. 19.7.6.

(23) VARGA - Kat. Sul. 3.7; Ap. Sul. 3.11; Hir. Sul. 1.50.

(24) VESTANA - Sankh. Gr. 3.I.8.

(25) VYASTI - Ap. Sr. I0.6; Saskh. Sr. 16.1.1.

(26) SUNYA - Drahy. Sr. 4.4.22; Nid. Su. 9.11.21.

(27) SESA - Kat. Sul. 3.2.3; Ap. Sul. 2.15; 13.2; Bau. Sul., 2.10.12.; Hir. Sul. 1.36; 4.34.

(28) SAMASTI - .Bau. Sr. 17.13.7; Jaim. Sr. 21.10.

(29) SAMYASAMUCCAYA - Kat. Sr. 1.8.7. 21,; 14. 3.5.

(30) SOPANTYA / UPANTYA - Ap. Sul. 1.12; Hir. Sul. 1.22.

This shows that most of the terms utilized. in the VM Sutras are not new but quite as old as the Srauta Sutras and the Sulba Sutras. In this connection it should be noted that the VM Sutras cannot be branded as `non-­Vedic' simply because they do not incorporate the very ancient archaic linguistic usages mostly found in the Vedic Samhitas or Brahmanas. In fact the Upanishads, which form the integral parts of some of the Samhitas and Brahmanas, are found to have been composed in a language which is more or less very similar to the one which is known as Paninian or Classical. And the Sulba Sutras too utilize the same sort of Sanskrit, as for instance,

Bau. Sul. 4.62-63         [Sanskrit line]

ibid., 7.6                       [Sanskrit line]

Man. Sul. 10.3.5.11     [Sanskrit line]

Ap. Sul. 3.21                [Sanskrit line]

ibid., 5.18                     [Sanskrit line]

ibid., 13.7-8                 [Sanskrit line]

Kat. Sul. 3.6-10           [four Sanskrit lines]

Thus, in point of the language, the VM Sutras too are similar and cannot be segregated as non-Vedic; they are as much Vedic as are the Srauta Sutras and the Sulba Sutras so far as the point of their language is concerned. And this is perhaps, because of their likelihood of being a part of the yet untraced Sulba-sutra of the Atharvaveda. And it is in view of these sutras being a part of the yet untraced Sulbasutra, and therefore belonging to the Sthapatyaveda an upaveda, of the Atharvaveda, that we may regard them as `Vedic', which is general term denoting not merely the texts connected with some Vedic Sakha, but not necessarily the Samhita and Brahmana only. Any text connected with the Veda Vidya or Vedic ritualistic, spiritual or any other practical aspect pertaining to the auxiliary sciences connected with Vedic Hinduism, may without hesitation be termed `Vedic'.

BKTM's Concept of the `Veda' and `Vedic':

... "From time to time and from place to place during the last five decades and more, we have been repeatedly pointing out that the Vedas (the most ancient scriptures, nay, the oldest `Religious' scriptures of the whole world) claim to deal with all braches of learning (spiritual and temporal) and to give the earnest seeker after knowledge all the requisite instructions and guidance in full detail and on scientifically - nay, mathematically, accurate lines in them all and so on."

"The very world `Veda' has this derivational meaning, i.e., the fountain-head and illimitable storehouse of all knowledge. This derivation, in effect, means, connotes and implies that the Vedas should contain within themselves all the knowledge needed by mankind relating not only to the so-called `spiritual' (or non-worldly) matters but also to those usually described as purely `secular', `temporal' or `worldly', and also to the means required by humanity as such for the achievement of all-round, complete and perfect success in all conceivable directions and that there can be no adjectival or restrictive epithet calculated (or tending) to limit that knowledge down in any sphere, any direction or any respect whatsoever."

"In other words, it connotes and implies that our ancient Indian Vedic lore should be all-round complete and perfect and able to throw the fullest necessary light on all the matters which any aspiring seeker after knowledge can possibly seek to be enlightened on."³⁶

As to the metaphysics, mathematics, physics and other subjects, BKTM has felt from a long time ago, that the Vedas which openly declare that these subjects are integral parts of the Vedic scriptures should be studied from the standpoint of the scholar, from the standpoint of a person who is absolutely impartial, has no prejudices, no presuppositions, a person who is a seeker after truth and welcomes the truth from whatever direction it may come. So we have to start with open minds, nothing a priory, nothing taken for granted. We have had long centuries of tradition over a long period of time, not merely various climes... So far as the Indian scriptures are concerned, we do not find any difficulty at all because each science is an integral part of a particular Veda, a particular scriptural portion of our literature.³⁷

As has been aptly remarked by Swami Pratyagatmananda Saraswati,³⁸ BKTM belonged to a race, now fast becoming extinct, of die-hard believers who think that the Vedas represent an inexhaustible mine of profoundest wisdom in matters both spiritual and temporal; and that this store of wisdom was not, as regards its assets of fundamental validity and value at least, gathered by laborious inductive and deductive methods of ordinary systematic inquiry, but was a direct gift of revelation to seers and sages who in their higher reaches of Yogic realisation were competent to receive it from a source, perfect, immaculate.

To carry conviction, BKTM has, by his comparative and critical study of the Vedic Mathematics, made abundantly clear the essential requirement of being prepared to go the whole length of testing and verification by accepted and accredited methods.³⁹

That there is a consolidated metaphysical background in the Vedas of the objective sciences including mathematics as regards the basic conceptions is a point that may be granted by a thinker who has looked broadly and deeply into both the realms.... That metaphysical background includes mathematics also; because physics as ever pursued is the application of mathematics to given or specified space-time-event situations. The late Shankaracharya has claimed, and rightly we may think, that the Vedic Sutras and their appli­cations possess these virtues to a degree of eminence that cannot be challenged. The outstanding merit of this work lies in his actual proving of this contention.⁴⁰

Whether or not the Vedas be believed as repositories of perfect wisdom, it is unquestionable that the Vedic race lived not merely as pastoral folk possessing half-or-quarter-developed culture and civilisation. The Vedic seers were, again, not mere `navel-gazers' or `nose-tip-gazers'. They proved themselves adepts in all levels and branches of knowledge, theoretical and practical. For example, they had their varied objective science, both pure and applied... The old seer scientist had both his own theory and art (technique) for producing the result, but different from those now prevailing. He had his science and technique, called Yajna, in which Mantra, Yantra and other factors must co-operate with mathematical determinateness and precision. For this purpose, he had developed the six auxiliaries of the Vedas in each of which mathematical skill and adroitness, occult or otherwise, play the decisive role. The sutras lay down the shortest and surest lines.⁴¹

In his Foreword of the General Editor, Dr. V. S. Agrawala has aptly pointed out⁴¹ that the question naturally arises whether the Sutras which form. the basis of this treatise, viz., the VM, exist any where in the Vedic literature as known to us. And we find that the Sutras in the form presented by BKTM, have not been found to form any part of the texts of the Atharvaveda Samhita, both Saunaka and Paippalada, nor the Gopatha Brahmana, nor the Kausika Sutra, nor the Vaitana Sutra, nor in any of the Upanisads traditionally belonging to the Atharvaveda, nor in any of the extent Sulba Sutras of both the Krsna and the Sukla Yajurveda Sakhas, nor in the Atharvaveda-parisistas.⁴²

But, says Dr. V. S. Agrawala, this criticism loses all its force if we inform ourselves of the definition of Veda given by BKTM himself as quoted of the definition of Veda given by BKTM himself as quoted above. It is the whole essence of his assessment of Vedic tradition that it is not to be approached from a factual standpoint but from the ideal standpoint, viz., that the Vedas as traditionally accepted in India are the repository of all knowledge, and hence they should be, and not what they are, in human possession. That approach entirely turns the tables on all the critics, for the authorship of Vedic mathematics then need not be laboriously searched in the texts as preserved from antiquity.⁴³

Even then, with due deference to the statements of a kind-hearted impartial scholarly saint like BKTM, we have yet to endeavour to trace the text of the `Ganita-sutra' or the Sulba-sutra, traditionally attached to the Atharvaveda, a part of which is evidently referred to by him. Till then, the chapter of the Vedic source of these Sutras cannot be finally closed.

Swami Pratyagatmananda Saraswati has referred to a consolidated metaphysical background in the Vedas of the objective sciences including mathematics as regards their basic conceptions.⁴⁴ It will, therefore, be useful to briefly survey the mathematical data as already available in the extant Vedic texts.

Mathematical Background in the Vedas :

Dr. R. P. Kulkarni has referred to the following data in the Vedas⁴⁵ with the remark that in fact the geometry of the Aryans is predominated by arithmetics, and that they have algebraised geometry.

The following figures with their increasing values occur in the Rgveda :

One year (1.110.4)                               Ninety four (1.155.6)

Two lips, Two breasts (2.39.6)             Ninety nine (1.54.6; 1.84.13)

Three wheeled chariot (3.34.2,5)          Hundred roads (1.36.16; 1.53.8)

Four eyes (1.31.13)                              Hundred and seven (10.97.1)

Five (1.164.12-13)                               150 (1.133.4)

Six horses (1.116.4)                             210 cows (8.19.37)

Seven Sindhus (7.35.8)                         300 cows (5.36.6)

Seven cities (1.6.7)                               360 spokes (1.164.48)

Eight directions (1.35.8)                        440 horses (8.55.3)

Nine days (1.116.24)                           720 (1.164.11)

Ten nights (1.116.24)                           1,000 ( 1.164.41 )

Eleven (1.139.11)                                 3,380 gods (3.9.9)

Twelve ( 1.164.48)                               4,000 (5.30.12)

Fourteen ( 10.114.6)                            9,000 (1.80.8)

Twenty Kings (1.53.9)              10,000 (5.27.1; 8.1.5)

Twenty one secret places (1.72.6)        30,000 (4.30.21)

Forty horses (1.126.4)                          60,000 cows ( 1.126.3)

Forty nine Maruts (7.51.3)                    60.099 (1.53.9)

Fifty maid servants (8.19.36)                70,000 (8.46.22)

Fifty three dice (10.34.8)                      One lakh (2.14.6)

Sixty three Maruts (8.96.8)                   Five lakhs (4.30.15)

Ninety (1.80.8; 1:121.13; 1.130.7.)      Many lakhs (2.14.7)

Series of two 2, 4, 6, 8 numbers of horses were used for chariot.

Series of ten 10, 20, 30, 40, 50, 60, 70, 80, 90, 100 (2.18.5-6).

The series of 10 upto 10¹² is given in the Yajurveda Samhita, with the successive terms as Eka (l), dasa (10), Sata (100), Sahasra (1,000), Ayuta (10,000), Niyuta (1,00,000), Prayuta (10,00,000), Arbuda (1,00,00,000), Nyarbuda (10,00,00,000), Samudra (1,00,00,00,000), Madhya (10,00,00,00,000), Anta (1,00,00,00,00,000) and Parardha (10,00,00,00,00,000). (Y.V.17.2)

The series of odd figures 1,3,5,7,9... 31,33 (Y.V.18.24).

The series of four 4,8,12,16... 48 (Y.V.18.25)

The series of 2 ,  1 2, 2 3, 3 and 4 (Y. V .18.26)

In the Taittiriya Samhita, too, the following series are mentioned (T.S.

8.2.11-20) :

The arithmetic series of odd figures 1,3,5,... 19, 29, 39... 99.

The series of two 2,4,6,8,... upto 20.

The series of four 4,8,12,16,20.

The series of five 5,10,15,20 ... upto 100.

The series of ten 10,20,30,40 ... upto 100.

The series of twenty 20,40,60 and 80.

The series of hundred 100,200,300 upto 1000.

The series of Ten from 10,100,1000... upto 10¹².

The multiplications : 4 × 5 = 100 = 5 × 20 = 10 × 10 = 20 × 5.

In the Atharvaveda large figures like Sata, Sahasra, Ayuta, and Nyarbuda are mentioned (Ath. V. 8.8.7).

The decimal system in the Atharveda is 5 and 50, 7 and 70, 9 and 90, 1 and 10, 2 and 20, 3 and 30 (A.V.6.25.1-3; 7.4.1; 19.4?.3-5).

The idea of infinity (ananta) is referred to in “[Sanskrit words]” (Y.V.16.54), “[Sanskrit words]” (Brh. Up. 2.5.19), “[Sanskrit words]” (Brh. Up. 3.2.12), and “[Sanskrit words]” (Brh. Up. 4.1.5).

There is a reference to the division of thousand by three in the Rgveda (6.69.8). That the division of thousand by three gives 333 × 3 + 1 is clearly mentioned in the Satapatha Brahmana (4.5.8.1). The fundamental operations given in this last work includes sums, multiplications, divisions and ratios.

Thus, we have sums like 12 + 5 + 3 + 1 = 21 (1.3.5.11); 10 + 10 + 10 + l = 31 (3.1.4.23); 3 + 3 + 4 = 10 (3.9.4.19); 10 + 10 + 2 + 2 = 24 (6.2.1.23); 12 + 5 + 3 + 1 = 21 (6.2.2.3); 10 + 4 + 1 + 1 + 1 = 17 (6.2.2.9); 24 + 12 = 36, 36 + 11 + 11 = 58, 58 = 60, 21 + 12 = 33, 33 + 11 + 11 = 55, 55 + 3 = 58, 58 + 2 = 60, 17 + 12 = 29, 29 + 1I + 11 = 51, 51 + 3 = 54, 54 + 2 + 2 = 58, 58 + 2 = 60 (6.2.2.31-37); 21 + 1 + 1 + 2 + 2 + 3 = 32 (7.1.2.16-17); 60 + 24 + 13 + 3 + 1 = 101 (10.2.6.1).

We have multiplications like 360 × 2 = 720, 720 × 80 = 57600 (2.3.4.19-20). 1 year = 360 nights = 720 days and nights = 10,800 Muhurtas; 1 Ksipra = 15 × 10, 800 Munurtas; 1 Etarha = 15 K~ipras; 1 Ida = 15 Etarhas; 1 Prana - 15 Ida; and also 1 Ana - 15 Nimesa, 1 Nimesa - 15 Lomagarta, 1 Lomagarta = 15 Svedayana, 1 Stoka = 15 Lomagarta (12:3.2.3-5).

[A series of numbers and fractions are given here. You will need to see the original article for this or request a copy from us]

The geometrical progression is mentioned in the Pancavimsa Brahmana as 12 , 24, 48, 96, 192, ... 3072, 6144, ... 49152, 98304, 196608, 393216 ( 18.3).

Yajurveda mentions the following fractions, viz., Tryavi, Tryavi, Dityavat, Difyauhi, Pancavi, Pancavi, Trivatsa, Trivatsa, Turyavat and Turyauhi (Y.V. 18.26).

[A series of numbers and fractions are given here. You will need to see the original article for this or request a copy from us]

Dr. S. N. Sen observes⁴⁶ that the method of obtaining higher and higher numbers in multiples of ten is clearly indicated. It appears that the thinking out and naming of such large numbers were a favourite pastime of the ancient Indian mathematicians. The Vedic Hindus showed the same proficiency in developing a scientific vocabulary of number names, in which the principles of addition, subtraction and multiplication were conveniently used. The system required the naming of the first nine digits, multiplying each of them by ten. The additive and the multiplicative principles are simultaneous used when, in the number concerned, the members from one to nine participate with multiples of ten.

 Acquaintance with the fundamental arithmetical operations with elementary fractions, progressive series is clearly indicated in the Vedic texts and their appendages.⁴⁷ The Vedic hymns make several references to arithmetical principles, like the consecutivity of numbers from 1 to 10 (AV. 13.4), additions of numbers with multiple of 10 (A.V. 5.15), additions of 2 (YV. 18.24), Additions of 4 (YV. 18.25), mention of the digit 99 (RV. 1.84.3), multiple by 11 (AV. 19.47), numeral system (YV. 17.2), and fractions.⁴⁹

The various Sulba-sutras that have been noted above have come down to us as parts of the Srauta-sutras and constitute Brahmanic geometrical manuals for the construction of sacrificial altars. In the various rules given, certain assumptions are taken for granted. Most of the postulates of the Sulba are concerning the division of figures, such as straight lines, rectangles, circles and triangles, and a few of them are about other matters of importance.⁴⁹ Of greater importance are the rules given in the Sulbas for the combination and transformation of rectilinear figures, specially the squares and the rectangles. The so-called `Pythagorean' theorem of the square of the diagonal is more explicitly given in more or less the same language in all the Sulbasutras we know of. There is hardly any doubt that the Vedic Sulbavids at this distant date possessed a valid proof of the theorem, of which the texts themselves provide reliable indications, while, as Junge has pointed out, the Greek literature of the first five centuries after Pythagoras contained no mention of the discovery of this or any other important geometrical theorem by the great philosopher and furthermore emphasized uncertainties in the statements of Plutarch and Proclus.⁵⁰ Regarding other areas of geometry, the area of a triangle, a parallelogram and a trapezium, as also the volume of a prism or cylinder and of the frustum of a pyramid are given.⁵¹ Another type of problem which interested the Sulba geometers was the circling of the square or its converse the squaring of the circle. Through these exercises, the Vedic Hindus were led to finding approximate values of.⁵²

The altar geometry of the Sulba-sutras does not fail to give us a glimpse of the beginnings of algebraic notions among the Vedic Hindus. The Quadratic Equation is utilized for the enlargement or reduction of the altar in accordance with a number of plans. The Sulbas contain rules for the construction of a square n times a given square; the rule involves the application of indeterminate equation of the second degree and simultaneous indeterminate equation of the first degree. Elementary operations with surds (karani) are clearly indicated in various places of the text. The Vedic Hindus have been credited further with the notion of irrationality of the quantities the square root of 2 and of 3.  The methods by which the values of these irrationals were obtained is not indicated in the texts; but more complete and clear statement and several indications of the derivations of approximate values are embedded in the very texts themselves. Another favourite mathematical pursuit of the Vedic Hindus was in the field of permutations, combinations. This interest was undoubtedly activated by the considerations of the Vedic meters and their variations. There were several Vedic meters with 6, 8, 9, 11, 12 syllables. The Vedic meter specialists were concerned with the problem of producing different possible types of meters from those of varying syllables by changing the long and short sounds within each syllable group. In this effort, they were led invariably to laying the foundation of the mathematics of permutation and combinations. Special importance attaches to Pingala's Chandah-sutras (200 B.C.) which contains a method called meru prastara for finding the number of combinations of n syllables taken 1, 2, 3, ... n at a time. The meru prastara is the same as the triangular array known in Europe as Pascal's triangle.⁵³

VM Terminology vis-a-vis that of the Medieval Mathematicians :

The medieval mathematicians like Aryabhata I (A.D. 476), Brahmagupta (A.D. 598), Bhaskara I (A.D. 600), Mahaviracarya (A.D. 850), Aryabhata II (A.D. 950), Sridharacarya (A.D. 991), Sripati (A.D. 1000), Bhaskara II (A.D. 1114), Narayana (A.D. 1350), and their commentators, have composed works like the Aryabhatiya, Brahmasphuta-siddhanta, Mahabhaskariya and Laghubhaskarlya, Ganita-sara-samgraha, Mahasiddhanta, Patiganita and Trisatika, Ganita-tilaka, Siddhantasiromani along with Lilavati and Bijaganita, and Ganitakaumudi, respectively. In these works they have utilized various technical terms for indicating various mathematical operations. Some of these terms they might have inherited traditionally, while some they might have newly coined.

The most important of the Hindu numberal notations is the decimal place-­value notation. In this system there are only ten symbols, those called anka (= mark) for the numbers one to nine, and the zero symbol, ordinarily called sunya (= empty).54 There is further a' system called Katapayadi in which the consonants of the Sanskrit alphabet have been used in the place of the numbers 1 to 9 and zero to express numbers; the conjoint vowels used in the formation have no numerical significance. It gives brief chronograms, which are generally pleasant sounding words. Scholars and scientists have tried to bring out astonishing data of scientific value from the application of this system to various Vedic texts.⁵⁵ BKTM has treated this topic in one full chapter in his VM.⁵⁶

The Hindu name for addition is Samkalita, with other equivalent terms like Samkalana, Misrana, Sammelana, Praksepana, Ekikarana, Yukti, Yoga and Abhyasa The word Samkalita has been used by some writers in the general sense of a series.⁵⁶ The terms Vyutkalita, Vyutkalana, Sodhana, Patana, Viyoga, etc., have been used for subtraction, while Sesa, Antara denote the remainder. The minuend is called Sarvadhana of Viyojya and the subtrahend Viyojaka.⁵⁷

The common Hindu name for multiplication is Gunana, which appears to be the oldest as it occurs in the Vedic literature. The terms Hanana, Vedha, Ksaya, etc. are also used for multiplication. The multiplicator is Gunya, multiplier is Gunaka or Gunakara, and product is Gunana-phala or Pratyutpanna.⁵⁹

Division seems to have been regarded as the inverse of multiplication; and the common Hindu names for the operation are Bhagahara, Bhajana, Harana, chedana, etc. The dividend is termed Bhajya or Harya, ete., the divisor Bhajaka, Bhagahara as simply Hara, and the quotient Labdhi or Labhdha.⁶⁰

The Sanskrit term for square is Varga or Krti⁶¹, while that for the cube is Ghana;⁶² and square-root is Varga-mula,⁶³ while the cube-root is Ghana­mula.⁶⁴. The symbols for powers and roots are abbreviations of Sanskrit words of those imports and are placed after the number affected. Thus, the square is represented by Va (from Varga), cube by Gha (from Ghana) the fouth power by Va-Va (from Varga-Varga), the fifth power by Va-Gha-Gha (from Varga-Ghana-Ghata), the sixth power by Gha-Va (from Ghana-Varga), the seventh power by Va-Va-Gha-Gha (from Varga-Varga-Ghana-Ghata) and so on. The product of two or more unknown quantities is indicated by writing Bha (from Bhavita) after the unknowns with or without the interposed dots; e.g. Yava-kagha-bha or Yavak'aghabha means (ya)³ (ka)³.⁶⁵ Brahmagupta mentions Varna as the symbols of unknowns,⁶⁵ generally represented by letters of the alphabet or by means of various colours such as Kalaka (Black), Nilaka (blue), Pitaka (yellow) and Haritaka (green). At one time the unknown quantity was called Yavat-Tavat (as many as, so much as) or its abbreviation Ya.⁶⁶ The multiplication of unknowns of unlike species is the same as the mutual product of symbols; it is called Bhavita.⁶⁸ The work of forming the equation is Samikarana, Samikriya or Samikara.⁶⁹ Writing down an equation for further operations is called Nyasa.⁷⁰ The operation to be performed on an equation next to is statement (nyasa) is technically known as Samassodhana, or simply Sodhana (= complete clearance), which varies according to the kind of the equation.⁷¹ Brahmagupta has classified equations as Eka-Varna-samikarana (equation with one unknown, anekavarna-samikarana (equations involving several unknowns), and Bhavita (equations involving products of unknowns); and the first class is again divided into two subclasses, viz., linear equations and quardratic equations. The principal of elimination of the middle term is Madhya­maharana.⁷² Ista-kanna is operation with an optional unknown, a method described by Bhaskara II,⁷³ One topic commonly discussed by almost all Hindu mathematicians goes by the special name of Samkramana (concurrence), Sankrama or Sankrama. Brahmagupta includes it in algebra while others consider it as falling within the scope of arithmetic; the subject of discussion here is the investigation of two quantities concurrent or grown together in the form of their sum and difference, in other words, the solution of the simultaneous equations x+y = a, x - y = b.⁷⁴ The process of solving the following two particular cases of simultaneous quardratic equations was distinguished by most Hindu mathematicians by the special designation Visama-krama (dissimilar operation) :

  ⁷⁵

 The subject of indeterminate analysis of the first degree is generally called by the Hindus Kuttaka, Kuttakara, Kuttikara or simply Kutta.⁷⁶ The equation by = ax + 1 is generally called by the name of Sthira-kuttaka or Drdha-kuttaka; later on the word Drdha was employed by later writers as equivalent of Niccheds (having no divisor) or Nirapavartya irreducible.⁷⁷ The indeterminate equations of the first degree put into the form

 by_l = a_lx  c_l

 by₂ = a₂x  c₂

 by₃ = a₃x  c₃

has, on account of its important applications in mathematical astronomy, received special treatment at the hands of Hindu algebraists from Aryabhat onwards. It is technically called Samslista-kuttaka.⁷⁸ The indeterminate quardratic equation Nx²  c = y² is called by the Hindus Varga-prakrti or Krtiprakrti (square-nature). and the absolute number or unity is Rupa, while the number which is associated with the square of the unknown is Prakrti. The number whose square, multiplied by an optional multiplier and then increased or decreased by another optional number, becomes capable of yielding a square-root, is designated by the term the lesser root-Kanistha-pada or Adya pada (first root). The root which results, after those operations have been performed, is called by the name Jyestha pada (greater root) or Anya­pada or Antya-pada (second root). If there. be a number multiplying both these roots, it is called the Udvartaka (augmenter); and, on the contrary, if there be a number dividing the roots, it is called the Apavartaka (abridger). The interpolator is Ksepaka.⁷⁹ Bhaskara II succeeded in evolving a very simple and elegant method by means of which one can derive an auxiliary equation having the required interpolator ±1, 2 or ±4, simultaneously with its two integral roots from another auxiliary equation empirically formed with any simple integral value of the interpolator, positive or negative; this method is called by the technical name Cakravala or the `Cyclic Method'.<sup>80</sup> Bhaskara II distinguishes two kinds of indeterminate equations as Sakrt­samikarana (Single Equation) and Asakrt-samikarana (Multiple Equation).<sup>81</sup> The indeterminate equation of the type bx + c = y<sup>2</sup> is called Varga-kuttaka or `Squarepulveriser',⁸² while that of the type bx + c = y³ is called the Ghana-kuttaka or `Cube-pulveriser'.⁸³

As compared to the above terminology of the medieval mathematicians, that of the VM is very simple and more akin to that of the Sulba-sutras, which are an integral part of the Kalpa Vedanga.

As has been rightly observed by Professors R. C. Gupta⁸⁴, the Sulba-sutras, although codified later on, represent much older material which was further enriched. Indian geometry on the Sulba-Sutras level could not have been created overnight, since the names of the altars for which it is utilised are as old as the Taittiriya Samhita and even the Rgveda. The Sulba-sutras material is not only important for mental and intellectual history of ancient India, but is also valuable for the history of human mind in general. It is also significant for investigating the philosophy, methodology and originality of Indian exact sciences.

The Methodology of the Medieval Mathematicians and that of BKTM in the VM

According the medieval mathematicians like Brahmagupta and other there are twenty operations (parikarma) and eight determinations (vyavahara) in Patiganita, i.e. calculation. The commentators have given the list of these logistics and determinations as follows :⁸⁵

(1) Samkaliram or addition; (2) Vyavakalitam or subtraction; (3) Gunana or multiplication; (4) Bhagahara or division; (5) Varga or square; (6) Varga­mula or square-root; (7) Ghana or cube; (8) Ghana-mula of note-book; (9-­13) Panca-jati or five standard forms of fraction; (14) Trairrasika or the rule of three; (15) Vyasta-rrairasika or the inverse rule of three; (16) Panca-rasika or the rule of five; (17) Sapta-rasika or the rule of seven; (18) Navarasika or the rule of nine; (19) Ekadasa-rasika or the rule of eleven; and (20) Bhanda-pratibhanda or barter and exchange; while the eight determinations are : (1) Misraka or mixture; (2) Sredhi or progression or series; (3) Ksetra or plane figures; (4) Khata or excavation; (5) Citi or stock; (6) Krakacika or saw; (7) Rasi or mound; and (8) Chaya or shadow.

That all mathematica1 operations are variations of two fundamental operations on addition and subtraction was recognised by the Hindu mathematicians, from early times.86 For addition there were two processes, viz., Direct and Inverse. For subtraction too there were both such processes. But BKTM has not touched these two operations, and he seems to have taken for granted the current prevalent methods in India, which includes mere mechanical application of a set of formulae committed to memory. BKTM has started with the multiplication for which he has utilized the `Urdhva-tiryak' sutra. Medieval mathematicians use five methods, viz., Gomutrika, Khanda, Bheda and Ista, as also the Kapata-sandhi. Datta and Singh have noticed seven distinct modes of multiplication employed by the Hindus, viz., Door junction Method, Gelosia Method, Cross Multiplication Method, Multiplication by Separation of Places, Zigzag Method, Parts Multiplication Method and Algennair Method.<sup>87</sup> Out of these the Cross Multiplication Method is algebraic and has been compared to Tiryak-gunana or Vajrabhyasa (cross multiplication) used in algebra.<sup>88</sup> This method was known to the Hindu scholars of the eight century, or earlier. BKTM's method is a simplied version of this method. Division was not considered to be difficult, as the most satisfactory method of performing it had been evolved at a very early period. In fact no Hindu mathematician seems to have attached any great importance to this operation, as it was considered to be too elementary.<sup>89</sup> A method of division by removing common factors seem to have been employed in India as mentioned in early Jain works (c. 160). The modern method of division is the Method of Long Division, invented in India about the 4th Century A.D., if not earlier.<sup>90</sup> BKTM utilises the `Nikhilam' sutra, the `Paravartya' sutra, and some others for division. There is no comparable method to this in the medieval writers.

As regards the problems of factorisation, equations, squaring or cubing, or square-roots or cube-roots, decimals and geometrical operations, I would leave the comparison to veteran mathematicians, rather than venture in the field not quite familiar to me as a Sanskritist.

Concluding Remarks :

In his recent article Dr. J. N. Kapur⁹¹ has observed that the word `all' in the sub-title `Sixteen Simple Mathematical Formulae from the Vedas, the one-line answers to all mathematical problems' has led to a great deal of confusion. The book, viz., the VM, deals with arithmetic and computation, but mathematics should not be confused with computation. In the minds of a great many people the terms `mathematics' and `computation' are synonymous. To be a `great mathematician' is to be a `rapid computer'. However, in some branches of mathematics, calculations play a very minor, even negligible, role. If a mathematician has to answer a question which calls for a number, he may have to do some computation to obtain the required result. However, the essential part of the solution in the problem is not the computation, but the reasoning process which enables the solver to choose the appropriate computation. It is this intellectual effort of analysing the solution that constitutes the mathematical character of the problem. And Prof. Kapur's quotation from Court, can be supported by a remark from Bhaskara who states in his Bija-ganita that the intelligent mathematicians should devise by their own sagacity all such artifices as will make the case for the method of the Square-nature and then determine the values of the unknowns.⁹²

As per the assessment of Prof. Kaput,⁹³ this book, the VM, is not concerned with those aspects of mathematics which do not depend on computation. However, most of the applicable parts of mathematics do require computation, but even here the real mathematical part is non-computational thinking and logical part, for which the present book does not provide any help. The book deals with only a small aspect of mathematics and its claim to give one-line answers to all mathematical problems is false, since most mathematical work in the VM consists of the following stages: doing experiments with numbers; recognising patterns and making conjectures; proving the conjectures; making these into theorems; , generalising the theorems; and abstracting the results to give them a larger degree of applicability; it deals with the first two aspects in the mathematical process, in fact mainly the second aspect only. Apparently Swamiji spent years in doing experiments with numbers and recognised patterns which could simplify the computation process, and then be followed the Vedic tradition by writing the formulae in very compact form. He also followed the Vedic tradition of not giving proofs. A formula may be given and verified a million times, yet it is not considered as a part of mathematics unless a rigorous proof is available. The Late Prof. P. L. Bhatnagar gave proofs of most of the formulae given by Swamiji⁹⁴ and the rest of the formulae can easily be proved. The proofs require only intermediate level of mathematics at which Swamiji worked and to regard it as high-level research in mathematics will again be making a false claim. Thus, according to Prof. Kapur, VM has nothing to do with mathematics in the Vedas except that it was written by a person who knew both Vedas and mathematics.⁹⁵ This remark is rather too much categorical, especially for a scholar like Prof. Kapur, who has been a veteran mathematician alright, but he does not know the Vedas first hand.

In my opinion, based on the first hand acquaintance with the contents of the Vedic texts referred to above, and with the VM as presented by BKTM, and in view of the scholarly, impartial, truth-loving and highly inquisitive nature and the versatile mind of the revered personality like him, and in view of the declarations he himself has made in his lecture in U.S.A., it seems to be beyond doubt that he did come across some Sulba-Sutra traditionally associated with the Atharvaveda, picked up some sixteen sutras initially, and later on some other thirteen or more sutras, worked on it for eight years from mathematical point of view, and in the light of his first hand knowledge of mathematics discovered the mathematical application of the sutras. In a way he may be called a mathematical interpreter of the `Ganita­sutras' which he happened to come across, but the real significance of which was long forgotten and could not be caught even by the commentators.

Similarly, the remarks of the Editorial Reviewer of International Dayananda Veda Peetha Research Journal,⁹⁶ that the Jagadguru has done positively a disservice to those who are interested in the history of ancient Indian Mathematics, is quite beside the point, and of no consequence. In his opinion, these sutras are entirely Jagadguru's imagination or intuitional visualization, or revealed to him personally. I think it is the mathematical interpretation of them that was actually revealed to him intuitively.

In one of his papers Shri `Ganitanand'⁹⁷ has remarked about the VM that the objectionable things about the book or system are the name "Vedic Mathematics" given to it and the claim that the 16 Formulas are from the Vedas, that both are deceptive and false and are responsible for creating lot of confusion and misunderstanding; the book or the sutras have nothing to do with the Vedas. This is also too much categorical for the scholar, since it will preclude the possibility of the new and scientific interpretation of old texts. The scholar can easily dismiss the newly propounded interpretations of a few Vedic texts by M. M. Pandit Madhusudan Oza, Dr. V. S. Aggrawal or Anand Coomarswamy as but figments of imagination, or having no more worth than that of a fiction. At the same time, he seems to agree with A. P. Nicholas by granting the fact that BKTM's system is the most delightful chapters of the 20th century mathematical history, and that the book has its own mathematical excellence or intrinsic importance. Its novelty has inspired several scholars to delve into the new area of research created by it and found more significant results through its approach. Its popularity is increasing both in India and outside.

Finally, we may agree with Dr. T. M. Karade⁹⁸ that you will never find a special and separate chapter on mathematics in the Vedas. It may be scattered here and there and to discover it is a difficult task indeed. However, one of the most successful attempts was made by BKTM in this direction. His work on mathematics is usually referred to as VM. But it does not mean that there cannot be VM other than that of BKTM.

END-NOTES :

l. Referred to as VM herein-after; published by Hindu Vishvavidyalaya, Sanskrit publications Board, Banaras Hindu Universiy, Varanasi, 1965; reprinted by Motilal Banarsidass, New Delhi, several times.

2. Author's Preface to VM, p. xix.

3. VM, My Beloved Gurudeva, pp. ix-x. 4. ibid., pp: i-xi.

5. ibid., pp. ii-v.

6. The seniormost surviving Trustee of the Maharshi Agricultural Research Institute (Ahmedabad Centre) Trust, Ahmedabad.

7. A Professor in the Kalabhavan Polytechnic, Vadodara.

8. Trivedi, Smt, Manjulaben - VM, `My Beloved Gurudeva', p. x.

9. ibid., p. x.

10. BKTM - VM, Author's Preface, pp. xix-xx.

11. BKTM - V. Meta., pp. 167-168.

12. ibid., p. 163.

13. ibid., p. 158.

14. ibid.

15. ibid., pp. 163-167.

16. ibid., pp. 163-164.

17. ibid., p. 164.

18. VM, p. 362; V. Meta., p. 165.

19. V. Meta., p. 164-165.

20. ibid., p. 166.

21. ibid., p. 167.

22. Colebrooke, H. T. - `Essays On the Religion and Philosophy of the Hindus'. London, 1856; p. 69.

23. Wilson, Hayman Horace - `Essays and Lectures on the Religion of the Hindus', Delhi, 1976.

24. `The Hymns of the Atharvaveda - Translated with a Popular Commentary', Vols. I-II, Benaras, 1916-17.

25. ibid., Vol. I. Preface, p. xiii.

26. Monier-Williams - Skt. - Eng. - Dict., p. 207, Col. 2.

27. [Sanskrit line] – (Trivednum Edn., p. 33), quoted by Durgamohan Bhattacharyya, in `Fun.Th. Ath., p. 24.

28. Battacharyya, Prof. Durgamohan, op. cit., pp. 23-24.

29. ibid., pp. 18-19.

30. Datta, Bibhutibusan, Sc. Sul., p. l.

31. Mah. Bh~s. on the Varttika "[Sanskrit line]" in the on I. i.i : [Sanskrit line] 

32. Datta, B., Sc. Sul., pp. 1-2.

33. ibid., pp. 6-7.

34. ibid., p.2.

35. VM., `Author's Preface, p.xiii'

36. Satyakama Verma - `Technical Term of Vedic Mathematics.'

37. V. Meta., pp. 162-163.

38. VM., Foreword, p. 11.

39. ibid.

40. ibid., pp. 13-14.

41. ibid., p. 14.

42. VM., General Editor's Foreword, p. 5.

43. ibid., pp. 5-6.

44. VM., Foreword, p. 11.

45. Kulkami, Dr. R. P. - Geom. Sul., pp. 26-27.

46. Bose, Sen, Subbarayappa - Con. Hist. Sc. Ind., pp. 141-142.

47. ibid., p. 143.

48. Mehta, D. D. - Pos. Sc. Ved., p. 110.

49. Bose, Sen, Subbarayappa, p. 149.

50. ibid.

51. ibid.

52. ibid.

53. ibid., pp. 155-157.

54. Datta and Singh - Hist. Hin. Math., Pt. I, p. 38.

55. ibid., p. 69.

56. VM., Chap. XXV : `The Vedic Numerical Code', pp. 194-195.

57. Datta and Singh - op. cit., Vol. I, p. 130.

58. ibid., p. 132.

59. ibid., p. 134.

60. ibid., p. 150.

61. ibid., p. 155.

62. ibid., p. 162.

63. ibid., p. 169.

64. ibid., p. 175.

65. ibid., Pt. II, p. 15.

66. ibid., p. 17, f.n. 2.

67. ibid., pp. 17-18.

68. ibid., p. 26.

69. ibid., p. 28.

70. ibid., p. 30.

71. ibid., p. 33.

72. ibid., p. 35. .

73. Bhaskara - `Lilavati', p. 76.

74. Datta and Singh - `Hist. Hin. Math.', Pt. II, pp. 43-44.

75. ibid., pp. 84-85.

76. ibid., pp. 89-90.

77. ibid., pp. 117 ff.

78. ibid., p. 135.

79. ibid., pp. 140-141.

80: ibid., pp. 161-162.

81. ibid., p. 207.

82. ibid., p. 251.

83. ibid., p. 254.

84. Gupta, Prof. R. C. - `Vedic Mathematics From the Sulva Sutras, Ind. Jour. Math. Edu., Vol. IX, March-July, 1989, pp. 2-3.

85. Datta and Singh - `Hist Hin. Math.', Pt.-I, p. 124.

86. ibid. pp. 129-130.

87. ibid., pp. 133-149.

88. Colebrooke, H. T. - `Hindu Algebra', p. 171, f.n. 5.

89. Datta and Singh - `Hist Hin. Math.; Pt. I, p. 146.

90. ibid., pp. 150-153.

91. Kapur, Prof. J. N. - `The So-called Vedic Mathematics', Mathematical Education, April-June, 1989, pp. 201-202.

92. Datta and Singh - `Hist. Hin. Math.', Pt. II, p. 183.

93. Kapur, J.N., op. cit., p. 202.

94. Bhatnagar, Prof. P.L. - Mathematics Teacher, Vol. x (1976), pp. 83-lll:

95. Kapur, J.N. - op. cit., p. 203.

96. Jomnal of the International Dayananda Veda Peetha, Vol. I, No. 1 (March 1988), New Delhi, pp. 123-129.

97. Ganitananda - `In the Name of Vedic Mathematics', Ganita Bharati, Bul. of Ind. Soc. for Hist. Math., Vol. 10, Nos. 1-4, 1988, p. 78.

98. Karade, Dr. T.M. - `A Word About Vedic Mathematics', Vaidic Ganita, Vol. I, 1985, Nagpur.

BIBLIOGRAPHY :

1. Ancient Indian Mathematics and Vedha by L. V. Gurjar, Poona, 1947.

2. Ancient Vedic Mathematics (Pushp - 3) by Dr. Narinder Puri: Publ.

Spiritual Science Series, Roorkee, 1989.

3. Aryabhatiya of Aryabhata (with Commentary of Bhaskara I and Someshvara). Edt. K. S. Shukla & K. V. Sharma. Publ. Indian National Science Academy, New Delhi, 1976.

4. Aryabhatlya of Aryabhaia (with Commentary of Bhaskara I and Someshvara). Edt. K. S. Shukla & K. V. Sharma. Publ. Indian National Science Academy, New Delhi, 1976.

5. Aryabhati'ya of Aryabhata (with Commentary of Suryadeva Yajvan). Edt. K. V. Sharma. Publ. Indian National Science Academy, New Delhi, I 976.

6. Asiatic Researches, Vol. III and VII, by H. T. Colebrooke.

7. Atharva Veda of the Paippaladas. Edt. Prof. Dr. Raghu Vira, Vol. I : Books 1-13. Publ. The International Academy of Indian Culture, Lahore, 1936.

8. Atharvaveda Samhita (Saunakiya). AV.

9. Basic Mathematics, by L. C. Jain & G. C. Patni. Publ. Rajasthan Prakrit Bharati Sansthan, Jaipur, 1982.

10. A Concise History of Science in India, by D. M. Bose, S. N. Sen and B. V. Subbarayappa. Publ. Indian National Science Academy, New Delhi, 1971.

11. Discover Vedic Mathematics, by Kenneth Williams, London, 1986.

12. Essays and lectures on the Religions of the Hindus, Vols. I-II, by H. H. Wilson. Publ. Asian Publication Service, New Delhi, 1976.

13. Essays on the Religion and Philosophy of the Hindus, by H. T. Colebrooke. Publ. Williams and Norgate, London, 1858.

14. The Fundamental Themes of the Atharvaveda, by Prof. Durgamohan Bhattacharyya. Publ. S. P. Mandali, Poona, 1968.

15. Ganita Bharati. Bulletin of the Indian Society for History of Mathematics, Tenth Anniversary Volume, Vo. 10, 1988, Nos. 1-4.

16. Geometry in Ancient India, by Svami Satya Prakash Sarasvati. Publ. Govindram Hasanand, Delhi, 1987.

17. Geometry According to Sulba Sutra, by Dr. R. P. Kulkarni. Publ. Vaidika Samshodhana Mandala, Pune, 1983.

18. A Glimpse into Vedic Mathematics (B.K.T. System), by Prin. V. G. Oke. Publ. Gokhale Education Society, Nashik.

19. Hymns of the Atharvaveda, Translated with a Popular Commentary, by Ralph T. H. Griffith, Vols. I and II. Publ. E. J. Lazarus & Co., Benares, 1916-17.

20. History of Hindu Mathematics, Parts I and II, by Bibhutibhushan Datta and Avdhesh Narayan Singh. Publ. Asia Publishing House, Bombay, 1962.

21. In the Name of Vedic Mathematics, by Ganitanand. Ganita Bharati, Bull.

Ind. Soc. Hist. Math., Vol: 10, Nos. 1-4, 1988, pp. 75-78.

22. Kanva-Samhita, of Sukla Yajurveda, Edt. Pandit Shripad Damodar Satavalekar. Publ. Svadhyay Mandal, Pardi, 1983.

23. Katyayana Sulba Sutra. Edt. by S. D. Khadilkar. Publ. Vaidika Samsodhana Mandal, Pune, 1974.

24. Kathaka Samhita of Yajurveda. Edt. by Pt. Shripad Damodar Satavalekar, Publ. Svadhyay Mandal, Pardi, 1983.

25. Kausika Sutra of Atharvaveda. Edt. Maurice Bloomfield. Publ. Motilal Banarsidass, Delhi, 1972.

26. Lectures on Vedic Mathematics, by A. P. Nicholas, J. Pickles and K. Williams, London, 1983.

27. Mathematics in Ancient India, by A. K. Bag: Publ. Chaukhamba. Orientalia, Varanasi, 1979.

28. Miscellaneous Essays, by H. T. Colebrooke, 'Second Edn., Vols. I-II. Publ. Higginobthem & Co. Madras, 1982.

29. Positive Sciences in the Vedas, by D. D. Mehta. Publ. Academy of Vedic Research, New Delhi, 1974.

30. Round Table Discussion on Vedic Mathematics, by Dr. Narinder Prui, 1989.

31. Rgveda Samhita. Edt. Pt. Shripad Damodar Satavalekar. Publ. Svadhyay Mandal, Pardi.

32. Siddhanta Siromani of Bhaskaracarya, by Dr. D. Arkasomayaji. Publ. Kendriya Sanskrit Vidyapeetha, Tirupati, 1980.

33. The Science of Sulba . A Study in Early Hindu Geometry, by Bibhutibhushan Datta. Publ. University of Calcutta, 1932.

34. The So-called Vedic Mathematics, by J. N. Kapur, Mathematical Education, April-June, 1989.

35. Samaveda Samhita. Edt. Pandit Shripad Damodar SatavaTekar. Publ. Svadhyay Mandal, Pardi.

36. Triples, by Kenneth Williams, London, 1984.

37. Taittiriya Samhita - Krsna yajurvediya, Edt. by Pt. Shripad Damodar Satavalekar. Publ Svadhyay Mandal, Pardi, 1983.

38. Vertically and Crosswise, by A. Nicholas, K. R. Williams and J. Pickles, London, 1984.

39. Vaidic Ganit. Publ. Einstein Foundation International, Vol. I, 1985, Vol. II, I 986.

40. Vaitana Sutra, Edt. by Richard Garbe. Publ. Mahalaxmi Publishing House, New Delhi, 1982.

41. Varaha Srauta Sutra, Edt. by Dr. W. Caland and Dr. Raghu Vira. Publ. Meharchand Lachhmandas, Delhi, 1971.

42. Vedic Metaphysics, by Jagadguru Shankaracharya Sri Sri Bharati Krishna Tirthaji Maharaja. Publ. Motilal Banarsidass, Deli, 1983:

43. Vedic Mathematics or Sixteen Simple Mathematical Formulae from the Vedas, by Jagadguru Swami Sri Bharati Krishna Tirthaji Maharaj, Publ. Hindu Vishvavidyalaya, Sanskrit Publication Board, Banaras Hindu University, 1965.

44. Workshop on Vedic Mathernatics (Ahmedabad), by Drs. Narinder Puri, P. S. Mishra, H. G. Sharma; G. S. Agarwal and Km. Pragati Goel, 1988.

45. Vedic Mathematics - A Review, by Editorial Reviewer. The Journal of

the International Dayananda Veda Peetha. Vol. I, No. l, March, 1988, New Delhi, pp. 123-129.

46. Vedic Mathematics from the Sulba Sutras, by Prof. R. C. Gupta. Indian Journal of Mathematical Education, Vol. 9, No. 2, July, 1989.

47. Workshop on Vedic Mathematics (Jaipur), by Dr. Narinder Puri, 1988.

48. Yajurveda Sarhhita - Vajasaneyi Madhyandina Sukla, Edt. Pt. Shripad Damodar Satavalekar. Publ. Svadhyaya Mandal, Pardi, 1970.