Vedic Mathematics | Pythagoras Theorem | Evolution of Trigonometry in India

In the area of Mathematical Contributions, India stands shoulder to shoulder as a sister civilization to any ancient civilization. It has received knowledge from other civilizations and given knowledge to other civilizations. However, not everyone would agree with this assessment. For some, vedas were depository of all knowledge including mathematics and we required nothing from other civilizations. In this article, Jerry Thomas examines the evidence presented for the claim that Vedas are the depository of all knowledge or so-called vedic mathematics. We will be also briefly examining the roots of Pythagoras theorem mentioned in Sulba Sutras, the development of zero as a digit in India and evolution of trigonometry to development of infinite series of trigonometric functions in Kerala School of Mathematics to demonstrate this point. This article is thus divided into the following sections:

• (A) Differing Views on India’s Contribution to Mathematics
• (B) Examining the Evidence for the Claim of Vedic Mathematics
• (C) Ancient Period: The Roots of Pythagoras Theorem in Sulba Sutras
• (D) Medieval Period: Development of Zero as a Digit
• (E) Medieval to Modern Times: Evolution of Trigonometry in India and Development of Infinite Series of Trigonometric Functions in Kerala School of Mathematics.
• 

Let us begin by looking at two differing views on India’s Contribution to Mathematics.

 

SECTION A: DIFFERING VIEWS ON INDIA’S CONTRIBUTION TO MATHEMATICS

Nobel Laureate Prof Amartya Sen, in his address to Infosys Award Ceremony in 2015, remarked this about the golden age of Indian Mathematics:

“…the golden age of Indian mathematics, which changed the face of mathematics in the world, was roughly from the fifth to the twelfth century, and its beginning was directly inspired by what we Indians were learning from work done in Babylon, Greece and Rome. To be sure there was an Indian tradition of analytical thinking, going back much further, on which the stellar outbursts of mathematical work in India from around the fifth century drew, but we learned a lot about theorems and proofs and rigorous mathematical reasoning from the Greeks and the Romans and the Babylonians. There is no shame in learning from others, and then putting what we have learned to good use, and going on to create new knowledge, new understanding, and thrillingly novel ideas and results…”

Source for full text: https://scroll.in/article/699603/golden-age-of-indian-mathematics-was-inspired-by-babylon-and-greece-amartya-sen

Anyone who has studied the Golden Age of Indian Mathematics would agree more or less with Prof. Amartya Sen.However, not everyone would agree with the assessment of Prof Amartya Sen. Among those who disagree, Swami Bharati Krishna Tirtha occupies a prominent position at least for two reasons – (1) Swami Bharati Krishna Tirtha (March 1884 – February 2, 1960), was the Shankaracharya of Govardhanamatha in Puri, (Odishā) from 1925 to 1960. In other words, he is not just another Hindutva braggard but a key leader of all Hindus. (2) He also wrote a book called as “Vedic Mathematics or Sixteen Simple Mathematical Formulae from the Vedas”, wherein Swami Bharti Krishna Tirthanot only claimed that Vedas contain all knowledge, whether spiritual or temporal but also mentioned a few sutras supposedly from Atharva Veda which could easily solve many of the mathematical problems which otherwise would take several steps. In fact, he was planning 16 volumes to illustrate how vedas apply to all branches of Mathematics. Let me quotea few of the points from preface of Vedic Mathematics written by Swami Bharati Krishna. The entire preface is worth reading to understand the perspective of Swami Bharti Krishna Tirtha.(For the full preface, please refer to the Appendix 1 – Swami Bharti Krishna Tirtha Preface).  The following verbatim excerpts from Swami Bharti Krishna Tirtha’s preface might help in grasping a few key relevant points. The numberings below are from his original preface:

2. The very word “Veda” has this derivational meaning i.e. the fountain-head and illimitable store-house 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 other-worldly) matters but also to those usually described as purely “secular”, “temporal”, or “wordly”; 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.
3. 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 matters which any aspiring seeker after knowledge can possibly seek to be enlightened on.

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

9. And we were agreeably astonished and intensely gratified to find that exceedingly tough mathematical problems (which the mathematically most advanced present day Western scientific world had spent huge lots of time, energy and money on and which even now it solves with the utmost difficulty and after vast labour involving large numbers of difficult, tedious and cumbeisome “steps” of working) can be easily and readily solved with the help of these ultra-easy Vedic Sutras (or mathematical aphorisms) contained in the Parisista (the Appendix portion) of the ATHARVAVEDA in a few simple steps and by methods which can be conscientiously described as mere “mental arithmetic”.
10. It is manifestly impossible, in the course of a short note (in the nature of a “ trailer” ), to give a full, detailed, thorough- going, comprehensive and exhaustive description of the unique features and startling characteristics of all the mathematical lore in question. This can and will be done in the subsequent volumes of this series (dealing seriatim and in extenso with all the various portions of all the various branches of mathematics).

 We may, however, at this point, draw the earnest attention of every one concerned to the following salient items thereof:—

(i) 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. In fact, there is no part of mathematics, pure or applied, which is beyond their jurisdiction

 (ii) The Sutras are easy to understand, easy to apply and easy to remember ; and the whole work can be truthfully summarised in one word “ mental” !

 (iii) Even as regards complex problems involving a good number of mathematical operations (consecutively or even simultaneously to be performed), the time taken by the Vedic method will be a third, a fourth, a tenth or even a much smaller fraction of the time required according to modern (i.e. current) Western methods;

*Only one volume has been bequeathed by His Holiness to posterity cf p. x above—General Editor.

While Swami Bharati Krishna Tirtha could not write 16 volumes (in fact, the first volume was published posthumously), we have at least one volume in our hands to examine the validity of his claims. Moreover, as Swami Bharati Krishna Tirtha was demonstrating his vedic method at various places even before the book was written, there were scholars who raised questions to him directly. Let us now examine the evidence presented for so-called Vedic mathematics in his book.

 

SECTION B: EXAMINING THE EVIDENCE FOR THE CLAIM OF VEDIC MATHEMATICS

 

We will begin by asking the most basic question – are there any references for the so-called sixteen simple mathematical formulae or at least sutras mentioned by Swami Bharati Krishna Tirtha in his book “Vedic Mathematics or Sixteen Simple Mathematical Formulae from the Vedas”. Interestingly, no one has been able to find the so-called sutras anywhere in Vedas. The so-called sutras seems to have been composed by Swami Bharati Krishna Tirtha attributing it to Yajur Veda.

In Myths and reality : On ‘Vedic mathematics’, Prof. ShrikrishnaGopalrao Dani (Prof. S G Dani hereafter), formerly at Tata Institute of Fundamental Research and currently Professor of Mathematics at Indian Institute of Technology, Bombay cites Prof. Kripa Shankar Shukla (Prof.  K S Shukla (1918 – 2007)) who retired from the Department of Mathematics and Astronomy of Lucknow University. Prof. K S Shukla, known for his contribution in bringing out landmark editions of twelve important source-works of Indian astronomy and mathematics, had once directly asked Swami Bharti Krishna Tirtha for the reference for sutras in Yajurveda but in vain. I quote Prof S G Dani:

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

It must be noted here that Swami Bharati Krishna Tirtha had ample of opportunity to produce evidence. If Prof K S Shukla asked for evidence in 1950, Swami Bharati Krishna Tirtha had 10 years to produce evidence before his death and the book was published in 1966. It is not merely that the sutras do not exist in Atharva Veda but also that they do not exist in any Sanskrit literature.

It is in this context that Prof Vasudeva Saran Agrawala, Sanskritist and a grammarian, the General Editor of the Vedic Mathematics or Sixteen Simple Mathematical Formulae from the Vedas gave a defense. In his Foreword to Vedic Mathematics or Sixteen Simple Mathematical Formulae from the Vedas 1965, Pages 5-6, he mentioned that while the sutras do not exist in any Vedic Literature, what Swami Bharati Krsna Tirtha meant was that it should have been part of Vedas and not that it was part of any existing literature then. I quote Prof Vasudeva Saran Agrawala:

“Swami Bharati Krsna Tirtha had the same reverential approach towards the Vedas. The question naturally arises as to whether the Sutras which form the basis of this treatise exist anywhere in the Vedic literature as known to us.

 But this criticism loses all its force if we inform ourselves of the definition of Veda given by Sri Sankaracarya himself as quoted below:

“Swami Bharati Krsna Tirtha had the same reverential approach towards the Vedas.  The very word ‘ Veda’ has this derivational meaning i.e. the fountainhead and illimitable store-house of all knowledge. This derivation, in effect, means, connotes and implies that the Vedas should contain (italics mine) within themselves all the knowledge needed by mankind relating not only to the socalled ‘ spiritual* (or other-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 (italics mine) all-round, complete and perfect and able to throw the fullest necessary light on all matters which any aspiring seeker after knowledge can possibly seek to be enlightened on”

In the light of the above definition and approach must be understood the author’s statement that the sixteen Sutras on which the present volume is based form part of a Parisista of the Atharva veda. We are aware that each Veda has its subsidiary apocryphal texts some of which remain in manuscripts and others have been printed but that formulation has not closed. For example, some Parisistas of the Atharva veda were edited by G. M. Bolling and J. VonNegelein, Liepzing, 1909-10. But this work of Sri Sankaracaryaji deserves to be regarded as a new Parisista by itself and it is not surprising that the Sutras mentioned herein do not appear in the hitherto known Parisistas.”

However, the so-called “”reverential approach towards the Vedas” defence by Prof Vasudeva Saran Agrawala carries no weight when we consider that Swami Bharti Krishna Tirtha’s entire attempt was to disprove Indologists who thought that there was nothing much in Vedas. I repeat the quote of Swami Bharti Krishna Tirtha:

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

 

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

How would writing a new appendix to Atharvaveda convincehistorians or for that matter anyone, that Vedas indeed had all knowledge from ancient times? Swami Bharti Krishna Tirtha’s work should be called as a fabricationto promote a pseudo-science (vedic science).

It must be also added that Swami Bharti Krishna Tirtha’s work is not even a summary or a derivation from any ancient Indian Mathematics.

I again quote  Myths and reality : On ‘Vedic mathematics’ by Prof. S G Dani.

“The contents of the swamiji’s book have practically nothing in common with what is known of the mathematics from the Vedic period or even with the subsequent rich tradition of mathematics in India until the advent of the modern era; incidentally, the descriptions of mathematical principles or procedures in ancient mathematical texts are quite explicit and not in terms of cryptic sutras. The very first chapter of the book (as also chapters XXVI to XXVIII) involves the notion of decimal fractions in an essential way. If the contents are to be Vedic, there would have had to be a good deal of familiarity with decimal fractions, even involving several digits, at that time. It turns out that while the Shulvasutras make extensive use of fractions in the usual form, nowhere is there any indication of fractions in decimal form. It is inconceivable that such an important notion would be left out, had it been known, from what are really like users manuals of those times, produced at different times over a prolonged period. Not only the Shulvasutras and the earlier Vedic works, but even the works of mathematicians such as Aryabhata, Brahmagupta and Bhaskara, are not found to contain any decimal fractions. Is it possible that none of them had access to some Vedic source that the swamiji could lay his hands on (and still not describe it specifically)? How far do we have to stretch our credulity? The fact is that the use of decimal fractions started only in the 16th century, propagated to a large extent by Francois Viete; the use of the decimal point (separating the integer and the fractional parts) itself, as a notation for the decimal representation, began only towards the end of the century and acquired popularity in the 17th century following their use in John Napier’s logarithm tables (see, for instance, Boyer, 1968, page 334).”

Further, the computational tricks in the so-called Vedic Mathematics follows a similar pattern of computational tricks in the Western Countries during those years. So much for Vedic Mathematics. To quoteMyths and reality : On ‘Vedic mathematics’ by Prof. S G Dani.

It may be recalled here that there have also been other instances of exposition and propagation of such faster methods of computation applicable in various special situations (without claims of their coming from ancient sources). Trachtenberg’s Speed System (see Arther and McShane, 1965) and Lester Meyers’ book, High-Speed Mathematics (Meyers, 1947) are some well-known examples of this. Trachtenberg had even set up an Institute in Germany to provide training in high-speed mathematics. While the swamiji’s methods are independent of these, for the most part they are similar in spirit. One may wonder why such methods are not commonly adopted for practical purposes. One main point is that they turn out to be quicker only for certain special classes of examples. For a general example the amount of effort involved (for instance, the count of the individual operations needed to be performed with digits, in arriving at the final answer) is about the same as required by the standard methods; in the swamiji’s book, this is often concealed by not writing some of the steps involved, viewing it as ”mental arithmetic.” Using such methods of fast arithmetic involves the ability or practice to recognise various patterns which would simplify the calculations. Without that, one would actually spend more time, in first trying to recognise patterns and then working by rote anyway, since in most cases it is not easy to find useful patterns.”

While Swami Bharti Krishna Tirtha’s Vedic Mathematics has nothing to do with Vedas and is just one of those many books of similar genre written in that time, it created much hype at the cost of real Indian Mathematics. Prof. S G Dani again notes in Myths and reality : On ‘Vedic mathematics’:

“For all their concern to inculcate a sense of national pride in children, those responsible for this have not cared for the simple fact that modern India has also produced several notable mathematicians and built a worthwhile edifice in mathematics (as also in many other areas). Harish Chandra’s work is held in great esteem all over the world and several leading seats of learning of our times pride themselves in having members pursuing his ideas; (see, for instance, Langlands, 1993). Even among those based in India, several like Syamdas Mukhopadhyay, Ganesh Prasad, B.N.Prasad, K.Anand Rau, T.Vijayaraghavan, S.S.Pillai, S.Minakshisundaram, Hansraj Gupta, K.G.Ramanathan, B.S.Madhava Rao, V.V.Narlikar, P.L.Bhatnagar and so on and also many living Indian mathematicians have carved a niche for themselves on the international mathematical scene (see Narasimhan, 1991). Ignoring all this while introducing the swamiji’s name in the ”brief history” would inevitably create a warped perspective in children’s minds, favouring gimmickry rather than professional work.”

Of course, facts hardly matter to those who want to promote their own narrow agendas. G. MadhavanNair, Former Chairman of ISRO, continues to perpuate this myth as late as in 2015. You can see his video on https://www.youtube.com/watch?v=6mL2Ix-amwQ (around 8:30 mins, he speaks about these 16 sutras as if those were ancient). However, for those who want more evidences are advised to read Prof. S G Dani’s papers, which are available at: http://www.math.tifr.res.in/~dani/

Now let us look at some of the real contributions beginning with the ancient period to the modern times.

 

SECTION C: ANCIENT PERIOD: THE ROOTS OF PYTHAGORAS THEOREM IN SULBA SUTRAS

 

In January 2015, at the 102 Indian Science Congress, then the Union Minister for Science and Technology, Dr Harsh Vardan said “Our scientists discovered the Pythagoras theorem but we very sophisticatedly gave its credit to the Greeks (Source: http://timesofindia.indiatimes.com/articleshow/45746060.cms?utm_source=contentofinterest&utm_medium=text&utm_campaign=cppst).

Dr. Shashi Tharoor backed Dr. Harsh Vardan and said “Modernists sneering at Harsh Vardhan should know he was right” (Source: https://indianexpress.com/article/india/india-others/tharoor-backs-harsh-vardhan-dont-debunk-ancient-science/).

So, what is the truth?

Dr.Manjul Bhargava, an Indian Origin Canadian-American mathematicianProfessor of Mathematics at Princeton University brilliantly summarized the story of  Pythagoras’s theorem in 2015 and  I quote:

“One is the one that originates in 2,500 BC in Egypt,” he adds. “There’s no statement of the theorem anywhere, but there is some knowledge that seems to be indicated of it because there are (Pythagorean) triples (when the length of the three sides are whole numbers, such as 34, and 5).”

According to Dr Bhargava, “The first systematic systems of listing a2 + b2 = c2 happens in the Plimpton tablets, which happen in about 1,800 BC (in Mesopotamia, or the modern-day Arab world). That shows a systematic understanding of producing solutions to that equation. That shows much more likelihood of knowledge of the Pythagorean theorem in a more general framework. But again there’s no written statement of the theorem.”

Arguing again that there are different standards for empirical evidence, Dr Bhargava says, “If you’re happy with a systematic solution to the equation that come up in the Pythagorean theorem even though there’s not a complete statement, one would say it came up with the Babylonians in Mesopotamia around 1,800 BC.”

“Another standard would involve,” he adds, “the requirement of a document that explicitly states the Pythagorean theorem — the geometric theorem. That first occurs about 800 BC in India in the Shuba Sutra of Baudhayan.”

“In that sense, if you want hard scientific evidence, it’s accurate to say that the Pythagorean theorem was first (recorded) in India in about 800 BC. Another standard could go beyond a mere statement,” says Dr Bhargava.

“One can go further than that — which is the standard that mathematicians often use — that while it’s nice to have the explicit statement but if there’s no proof well, then, maybe they (the ancient culture being studied) didn’t know it,” he says.

“The Shuba Sutras do contain proofs in some special cases and contain numerical proofs in general, but the first actual rigorous proof of the Pythagorean theorem that’s on record originates in China — after the Shuba Sutra.”

“So in China in school textbooks they often call it the Gougu theorem. And that was first given in a Chinese manuscript some years later (the Zhou Bi Suan Jing, the material for which dates back to sometime between the 1046 BC and 256 BC).”

“So maybe the statement of the theorem went from India to China, but the actual proof — the complete, rigorous proof — was given in China, at least as far as written records go. That’s why the Chinese … (named) the Pythagorean theorem after the person who first proved it (and) who was in China.”

Source: https://www.rediff.com/news/special/did-india-discover-pythogoras-theorem-a-top-mathematician-answers/20150109.htm

Dr.Manjul Bhargava’s apt summarization is supported by historians of mathematics.

In the Journal Historia Mathematica 7 (1980), 186-187, Beatrice Lumpkin writes:

“Richard J Gillings’ Mathematics in the Time of Pharaohs (1975, Cambridge, MIT Press), contains two clues which strongly suggest that the ancient Egyptians knew specific cases of the Pythagorean Theorem. Although he is careful to make no such claims, Gillings analyses two problems from the Berlin papyrus which suggest this is the case. They are listed in the index of his book under “Pythagorean theorem, suggestions of, in equations, page 161.”

While Richard J Gillings may not be explicit in Egyptian Berlin Papyrus, he is explicit in acknowledging the Pythagorean theorem in Plimpton Tablets..

In Mathematics in the Time of Pharaohs, Introduction, Cambridge, MIT Press, Richard I. Gillings, states that:

A second oddity of the history of mathematics was brought to light when the Babylonian clay tablet Plimpton 322 (museum number, Columbia University, New York) was translated by Neugebauer and Sachs in 1945. The translation established beyond any doubt that the Pythagorean theorem was well known to Babylonian mathematicians more than a thousand years before Pythagoras was born. The history books tell us that the Greek mathematician sacrificed an ox to celebrate the discovery of the theorem named after him. Here then is an unrewarded anticipation, for doubtless the name of the famous theorem will remain as a true mumpsimus-” the Pythagorean theorem”-for all time.

So, there you are – depending up on your standard of writing history, you can attribute it to Egypt, Babylon, India or China.

At this point one must also note that while Swami Bharati Krishna Tirtha decries Indologists, it was Dr. George Thibaut, an Indologist who studied Sulba Sutras, the first to highlight phythogras theorem in Sulba Sutras. He published his detailed paper in the Journal Of The Asiatic Society Of Bengal 1875 which was also published in the translation of Sulba Sutras of Baudhayana. Dr. George Thibaut pointed to the Phythogras theorem in Sulba Sutras. He wrote:

These propositions are as follows :

Baudhayana:

The cord, which is stretched across—in the diagonal of—a square produces an area of double the size. That is : the square of the diagonal of a square is twice as large as that square.

This is different from the way we studied Phyothgras theorem (which is in triangle) but it is essentially same.

So, what are Sulba Sutras? Are the mathematical texts or scientific papers of ancient Indian scientists? No. Those are texts for the construction of altars for some vedic rituals.

In A History of Indian Literature, Volume 4, Jyothisastra Astral and Mathematical Literature, Chapter 1, Sulba Sutras, David Pingree writes:

In the performance of Vedic srauta rituals an essential prerequisite is the piling up of the fire altar (agnicayana). These altars (citis) take the form of various objects; the forms mentioned in Taittirryasamhita 5, 4, 11 (after the chandasciti1 or “meter altar”) and the sacrificers who should erect them are:

1. Syenaciti or “hawk altar”2 by one desiring heaven (suvarga);
2. kankaciti or “heron altar” by one desiring a head in the other world;
3. alajaciti or “alaja-hird altar” with four furrows by one desiring support;
4. praiigaciti or “triangle altar” by one desiring to repel his foes;
5. ubhayatahpraiigaciti or “triangle on both sides altar” by one desiring to repel both present and future foes;
6. rathacakraciti or “chariot-wheel altar” by one wishing to defeat his foes;
7. dronaciti or “trough altar” by one desiring food;
8. samuhyaciti or “things to be gathered together altar” by one desiring cattle;
9. paricdyyaciti* or “circle altar” by one desiring a village;
10. dniasdnaciti* or “cemetery altar” by one desiring the world of the fathers (pitrloka).

A few other altar-shapes are described in other Brahmanas, where also are prescribed the rituals to be performed at these altars. The Srautasutras belonging to the Yajurveda often include as appendices treatises that give rules concerning the geometry involved in the construction of these altars. These treatises are known as the Sulbasutras.

Further, the construction of these different altars had to follow certain specifications. David Pingree adds:

Each of the basic altars must be constructed with five layers of bricks, and there must be a fixed number of bricks in each layer; moreover, the bricks in the second and fourth layers must not be directly above or below those in the first, third, and fifth layers. And the surface covered by the altar, regardless of its shape, must cover an area of seven and one half square purusas or, for certain purposes, that area increased by specified numbers of square purusas, or it must be multiplied by a given factor. Finally, the altar must be correctly oriented with respect to the cardinal directions. The task faced by the authors of the Sulbasutras was to prescribe rules for laying out these altars with only a rope (rajju or sulba) of determined length and posts or gnomons (sanku). The geometrical problems that were solved by these altar-builders are indeed impressive, but it would be a mistake to see in their works the unique origin of geometry;  others in India and elsewhere, whether in response to practical or theoretical problems, may well have advanced as far without their solutions having been committed to memory or eventually transcribed in manuscripts.

Sulba Sutras aided in the construction of vedic altars. While we are not interested in the vedic rituals or altars, we are interested in the construction technique of the altars.

Dr. George Thibaut himself made a distinction between the beliefs and the technique of construction. In his paper in the Journal Of The Asiatic Society Of Bengal 1875, he wrote:

But the chief interest of the matter does not lie in the superstitions fancies in which the wish of varying the shape of the altars may have originated, but in the geometrical operations without which these variations could not be accomplished. The old yajnikas had fixed for the most primitive chiti, the chaturasrasyenachit, an area of seven and a half square purushas, that means seven and a half squares, the side of which was equal to a purusha.

There has been much discussion on Sulba Sutras thereafter. One of the questions raised was whether the people who prescribed the vedic rituals are the same people who prescribed the techniques of construction methods as internal evidence indicates otherwise.

In 1875, Dr. George Thibaut did not have anyone else to cite as Indus Valley Civilization or later archaeological evidence were yet to unearthed. Let us look at some of the discussions around it.

In History of Science & Technology in Ancient India,  Chapter 5, Mathematics in its Making, DebiprasadChattopadhyaya points out:

Notwithstanding the usual assumption that this mathematics was -created by the Vedic priests., the internal evidences of the texts indicate that it was the direct outcome of the theoretical requirements mainly of the brick-makers and bricklayers, who were using burnt bricks and whose status within the general framework of the social norm of the Vedic priests is at best questionable.

 

Apart from the social question as to whether the vedic priest would involve in such brick construction, it must be noted that Sulba Sutras themselves refers to others when it comes to the technical part of construction.

In History of Science & Technology in Ancient India,  Chapter6, Technicians and the Vedic Priests, DebiprasadChattopadhyayacites BB Datta  who apparently subscribes to the most orthodox vedic beliefs:

What should be particularly emphasized now is the fact that those specifications are not due to the authors of the Sulba themselves. They do not even pretend to make any such claim. On the other hand, they have often and then expressly admitted to have taken them from earlier works. We, in fact, find that numerous passages of Baudhayana and ApastambaSulba dealing with the spatial magnitudes of sacrificial alta” as well as with the methods of their construction, end with the remark itivijnayate [or ‘it is known’, ‘it is recognised or prescribed (by authorities)]. Sometimes itiabhyupadisanti (‘thus they teach’) or iti “kram (‘it has been said’), is used in the same sense. It has been rightly pointed out before by Garbe that all those passages of Apastambaar~ literal quotations from the TaittiriyaBrahnuma or from the BrahmalUl-like portions of the TaitliriyaSamhIla or Aranyaka. That is exactly true also of the similar passages of Baudhayana. This writer is occasionally more explicit about his sources.

In addition to the fact that the authors of Sulba Sutra cite other authorities when it comes to the technical part, there is also some linguistic evidence. While the title of the Sutra is Sulba meaning rope in Sanskrit, the word Sulba is not used within the text. The word for rope used within the text is rajju, a local word which would have been used by craftsman but not the priests. In History of Science & Technology in Ancient India,  Chapter6, Technicians and the Vedic Priests, DebiprasadChattopadhyayaadds:

The very word Sulva for rope or cord is so elitist-esoteric that in the whole range of Sanskrit literature it is difficult to come across it outside the titles of the texts. Within the texts, however, the word is never used, the cord or rope being referred to as rajju. This peculiarity of the texts is already noted by Thibaut who observes: I may remark at once that the sutra-s themselves do not make use of the term sulva; a cord is regularly called by them rajju.” What needs to be added to it is only a simple point. The ward rajju is in fact so plebian that it is the same also in Pali. A craftsman would have been easily familiar with it and use it as part of his own vocabulary rather than the word sulva.. which was presumably current only among the priestly elites.

To add to all these is the interesting point that this very ritual was not part of the Rig Veda but evolved later in Yajurveda. In the paper Greek and Vedic Geometry, Frits Staal writes:

The only two Vedas that concern us are the Rig and Yajurveda. The Rigveda was composed during the second millenium B.C. (the bulk of it before 1100 B.C.: Witzel 1992: 614); but it does not mention the Agnicayana and refers to altars only thrice. In each case, the altar seems to be simple and part of the domestic ritual, performed within the home and different from the Agnicayana ritual which is a (relatively) “public” ritual. In the one (late) case in which the Rigveda description implies a shape, it is quadrangular (RV 10.114.3; cf. Potdar 1953: 73). In the domestic ritual of a few centuries later, about which more evidence is available, the altar is circular. Though not referred to, it is likely that this circular altar was already known at the time of the Rigveda.

 

The Yajurveda was composed between 1000 and 800 B.C. and a full third of it is devoted to the Agnicayana. The language is still similar to that of the Rigveda but begins to develop in different directions (see Renou 1956: Ch. I; Witzel 1989). Whatever the difference in language between the two Vedas with their numerous subdivisions, it is likely that there were also social and ethnic differences among its users. Most of the Rigveda was probably composed by members of the Seminomadic Indo European pastoralists whose tribes and clans had trickled in across the mountain ranges that separate Central Asia from Iran and the Indian subcontinent; whereas parts of the Yajurveda are more likely to have been composed by indigenous Indians who had become bilingual by adopting the language of these incoming tribes as a second language (cf. Deshpande 1993). Why they adopted this alien language has not been satisfactorily explained; but the fact is notin doubt and not confined to India: it holds true of IndoEuropean in general (see Mallory 1989: 257–261).

 

Or to rephrase, the oldest Veda, Rig Veda did not have this ritual and it has evolved over a period of time. So, now the question is who are the real authors of the technical aspect of Sulba Sutras. There are at least two possible responses – they could have been Indus Valley Inhabitants who knew brick technology exceedingly well who were definitely there when the Aryans migrated to India.

Frits Staaladds in his paper Greek and Vedic Geometry:

I have already referred to these MargianBactrian or BactroMargian peoples which are well known to archaeologists, especially Russian archaeologists, who made numerous excavations at this “BactrianMargianian Archaeological Complex” (BMAC). Though the BMAC civilization lasted only a few hundred years, between 2000 and 1500 B.C., much recent information is available on the abundant traces of its influence in the IndoIranian borderlands. According to Hiebert (1995: 199), “it is tempting to see the spread of the BMAC to Iran and South Asia as the diffusion of a new type of politicaleconomic and linguistic structure carried by a small group of people and adapted by local populations” (see also Hiebert and LambergKarlovsky 1994: 12).

I shall not try to pursue the archaeological ramifications of these discoveries which are numerous, but provide one illustration. Figure 6 provides the reconstruction by Viktor Sarianidi (1987: 52) of the Togolok temple, dated to about 1800–1500 B.C., a large rectangular complex (130 by 100 meters) with round corner towers and in one corner, on a plot especially reserved for them, two round brickfaced altars dug into the earth.

I believe we are now beginning to see the first outline of a demonstration of our hypothesis regarding the geographical area of origin of Vedic and Greek geometry. It is only a beginning, but it points in the right direction, and archaeologists have to take it from there. One more question arises. Since the techniques of firing bricks was known to the Indus Valley Harappans, were they also linked to this BMAC complex? This question has to a large extent been answered  by French archaeologists, especially Jean-Francois Jarrige, who has established the existence of extensive cultural exchanges between the two areas starting during the end of the third and continuing through the beginnings of the second millenium B.C. (see, e.g., Jarrige 1985 and cf. Parpola 1993: 47–48).

In order to summarize our discussion on Pythagoras theorem:

• Pythagorean Triples were known to Egyptians, the first systematic systems of listing a2 + b2 = c2 happens in the Plimpton tablets (around 1,800 BC) much before even Rig Veda was composed, the first record of theorem is in BaudhayanaSulba Sutra, and the proof of Pythagoras theorem was given by Chinese in Gougu theorem.
• Further, while Pythagoras theorem was recorded in an appendix of vedic texts, it is questionable whether even that knowledge was known to vedic people or it was borrowed from others.
• If anyone has any doubts whether a vedic limb would borrow from outside, please refer to the article published on astrology (http://www.sakshitimes.net/blog/2020/10/18/vedic-limb-indian-astrology-derived-paul-greeks-roman-vedic-golden-age-myth-jerry-thomas/)

SECTION D: MEDIEVAL PERIOD: DEVELOPMENT OF ZERO AS A DIGIT

 

Without exaggeration, one can say that the most important contribution that India gave to the world in Mathematics is the digit zero.

Swami Bharti Krishna Tirtha’s preface to Vedic Mathematics without doubt cites it and I quote:

16. The following few excerpts from the published writings of some universally acknowledged authorities in the domain of the history of mathematics, will speak eloquently for themselves:—

 

(i) On page 20 of his book “ On the Foundation and Technique of Arithmetic” , we find Prof. G.P. Halstead saying “ The importance of the creation of the zero mark can never be exaggerated. This giving of airy nothing not merely a local habitation and a name, a picture but helpful power is the characteristic of the Hindu race whence it sprang. It is like coining the Nirvana into dynamos. No single mathematical creation has been more potent for the general on-go of intelligence and power”.

 

However, there is much history to the digit zero.In the paper A history of Zero, by J J O’Connor and E F Robertson, School of Mathematics and Statistics, University of St Andrews, Scotland, provides a detailed and latest research on this subject. J J O’Connor and E F Robertson’s MacTutor History of Mathematics received Hirst Prize from London Mathematical Society. Prof E F Robertson also was elected as a Fellow of the Royal Society of Edinburgh.

I am quoting extensively from this paper and in fact is worth reading completely here (https://mathshistory.st-andrews.ac.uk/HistTopics/Zero/).

Initially, in the paper A history of Zero,J J O’Connor and E F Robertson tells us about the two uses of zero:

The first thing to say about zero is that there are two uses of zero which are both extremely important but are somewhat different. One use is as an empty place indicator in our place-value number system. Hence in a number like 2106 the zero is used so that the positions of the 2 and 1 are correct. Clearly 216 means something quite different. The second use of zero is as a number itself in the form we use it as 0. There are also different aspects of zero within these two uses, namely the concept, the notation, and the name. (Our name “zero” derives ultimately from the Arabic sifr which also gives us the word “cipher”.).

A word must be added here about Brahmi Numerals lest we think that we always had what we calls as Hindu Arab Numerals. In the paper, Indian numerals, J J O’Connor and E F Robertson remarks:

There were separate Brahmi symbols for 4, 5, 6, 7, 8, 9 but there were also symbols for 10, 100, 1000, … as well as 20, 30, 40, … , 90 and 200, 300, 400, …, 900.

The Brahmi numerals have been found in inscriptions in caves and on coins in regions near Poona, Bombay, and Uttar Pradesh. Dating these numerals tells us that they were in use over quite a long time span up to the 4th century AD. Of course different inscriptions differ somewhat in the style of the symbols.

In the paper, A history of Zero, J J O’Connor and E F Robertson describes the evolution of empty place holder for zero in Babylonian Mathematics around BC 700- 400:

One might think that once a place-value number system came into existence then the 0 as an empty place indicator is a necessary idea, yet the Babylonians had a place-value number system without this feature for over 1000 years. Moreover there is absolutely no evidence that the Babylonians felt that there was any problem with the ambiguity which existed. Remarkably, original texts survive from the era of Babylonian mathematics. The Babylonians wrote on tablets of unbaked clay, using cuneiform writing. The symbols were pressed into soft clay tablets with the slanted edge of a stylus and so had a wedge-shaped appearance (and hence the name cuneiform). Many tablets from around 1700 BC survive and we can read the original texts. Of course their notation for numbers was quite different from ours (and not based on 10 but on 60) but to translate into our notation they would not distinguish between 2106 and 216 (the context would have to show which was intended). It was not until around 400 BC that the Babylonians put two wedge symbols into the place where we would put zero to indicate which was meant, 216 or 21 ” 6.

The two wedges were not the only notation used, however, and on a tablet found at Kish, an ancient Mesopotamian city located east of Babylon in what is today south-central Iraq, a different notation is used. This tablet, thought to date from around 700 BC, uses three hooks to denote an empty place in the positional notation. Other tablets dated from around the same time use a single hook for an empty place. There is one common feature to this use of different marks to denote an empty position. This is the fact that it never occured at the end of the digits but always between two digits. So although we find 21 ” 6 we never find 216 ”. One has to assume that the older feeling that the context was sufficient to indicate which was intended still applied in these cases.

While Babylonians had used wedge as a place holder for zero, their base system was 60 (sexagesimal) and not 10 (decimal). However, China had decimal system with a vacant space for zero in their bar counting system – the Han counting-board from around AD 300:

In the book Science and Civilisation in China, Volume 3, Mathematics and the Sciences of the Heavens and the Earth, 19 Mathematics, Pages 148-149, the eminent scholar Joseph Needham writes:

But if China conserved with completeness her own characteristics style in mathematics until the beginning of the modern period, if India was the more receptive of the two cultures, one illustrious invention there was which seems to have occurred in the very border marches of the two great civilizations, into both of which it quickly spread. This was nothing less than development of a specific written symbol for nil value, emptiness, sunya, i.e. the zero. Perhaps we may venture to see in it an Indian garland thrown around the nothingness of the vacant space on the Han counting-board.                       

This raises, however, the question of the origin and peregrinations of that  fundamental manifestation of mathematical order, place-value of figures. Here the position of China is now fairly clear. In the Shang period (late – 2nd millennium) the system of writing numbers was more advanced and scientific than in any other culture of high antiquity. The nine digits appeared in all units, characterised at higher levels by place-value components. These components were not themselves numerals, but in the course of time conducted, as it were, the nine digits to their proper places in the columns of  the counting-boards on which calculations were made. The existence of such counting boards is implicit in all that we know of Chhin and Han mathematics, and it was quite natural that the absence of a digit at any unit level should be represented by an empty space. That this system was already complete about the – 4th century, the Mo Ching and the coin inscriptions testify.

 

In view of what has already been said, the derivation of the place-value principle in

China from that form of it which the Old Babylonian astronomers (early -2nd millennium) were the first to use, seems doubtful. Many principles of numeration

(additive, subtractive, multiplicative) were current in ancient Mesopotamia, and even

on the mathematical and astronomical cuneiform tablets, the sexagesimal place-value arrangement was combined with other principles for values lower than 60. Yet place-value in China from the Shang to the Han and onwards was decimal, not sexagesimal.

 

And with one slight exception it was not combined with any other principle. Since therefore in other fields strong evidence of transmission of techniques and inventions

from ancient Mesopotamia to China is noted,  we might perhaps regard the conception of place-value as a case of stimulus diffusion; the travel of a bare idea and not its concrete implementation.

 

But when we consider Indian developments, the conviction grows that the appreciation and use of the place-value principle was much posterior to its appearance in China. It cannot be traced in India with certainty before the early years of the + 5th century, the date of the PaulisaSiddhanta, though by the time of Aryabhata (+499) and his contemporaries such as Varaha-Mihira (author of the Panca-Siddhantika) it was without doubt employed. Significantly, it was the decimal place-value of China, not the sexagesimal place-value of ancient Babylonia. Then, towards the end of the + 7th century in Indo-China, and in the + 8th and + 9th in India itself, the inscriptions were made which attest the existence of the zero symbol, both in its dot and space–enclosing forms. These simply confirm that by then the place-value of figures was well understood. Now the period in question (+300 to +900) was approximately that to which we must ascribe many of the mathematical transmissions from China just mentioned, and it was also that which saw the great expansion of Buddhism in the Chinese culture-area. Could it be that the travelling monks exchanged mathematics for Indian metaphysics? An epic story of culture-contact may well await excavation  from the monastic biographies of the Kao Seng Chuan. We probably know only a few fragments of it, such as the fact, for instance, already referred to, c that about +440 a monk of the Northern Wei, Than-Ying, could be a distinguished teacher of the mathematics of the Chiu Chang.         

 

If Babylonians had a wedge as a place holder for zero, and then Chinese had decimal system with a vacant space for zero in their counting rod system, what exactly did we create?

J J O’Connor and E F Robertson writes in their A history of Zero:

What is certain is that by around 650AD the use of zero as a number came into Indian mathematics. The Indians also used a place-value system and zero was used to denote an empty place. In fact there is evidence of an empty place holder in positional numbers from as early as 200AD in India but some historians dismiss these as later forgeries. Let us examine this latter use first since it continues the development described above.

In around 500AD Aryabhata devised a number system which has no zero yet was a positional system. He used the word “kha” for position and it would be used later as the name for zero. There is evidence that a dot had been used in earlier Indian manuscripts to denote an empty place in positional notation. It is interesting that the same documents sometimes also used a dot to denote an unknown where we might use xx. Later Indian mathematicians had names for zero in positional numbers yet had no symbol for it. The first record of the Indian use of zero which is dated and agreed by all to be genuine was written in 876.

We have an inscription on a stone tablet which contains a date which translates to 876. The inscription concerns the town of Gwalior, 400 km south of Delhi, where they planted a garden 187 by 270 hastas which would produce enough flowers to allow 50 garlands per day to be given to the local temple. Both of the numbers 270 and 50 are denoted almost as they appear today although the 0 is smaller and slightly raised.

We now come to considering the first appearance of zero as a number. Let us first note that it is not in any sense a natural candidate for a number. From early times numbers are words which refer to collections of objects. Certainly the idea of number became more and more abstract and this abstraction then makes possible the consideration of zero and negative numbers which do not arise as properties of collections of objects. Of course the problem which arises when one tries to consider zero and negatives as numbers is how they interact in regard to the operations of arithmetic, addition, subtraction, multiplication and division. In three important books the Indian mathematicians Brahmagupta, Mahavira and Bhaskara tried to answer these questions.

Brahmagupta attempted to give the rules for arithmetic involving zero and negative numbers in the seventh century. He explained that given a number then if you subtract it from itself you obtain zero. He gave the following rules for addition which involve zero:-

The sum of zero and a negative number is negative, the sum of a positive number and zero is positive, the sum of zero and zero is zero.

Subtraction is a little harder:-

A negative number subtracted from zero is positive, a positive number subtracted from zero is negative, zero subtracted from a negative number is negative, zero subtracted from a positive number is positive, zero subtracted from zero is zero.

Brahmagupta then says that any number when multiplied by zero is zero but struggles when it comes to division:-

A positive or negative number when divided by zero is a fraction with the zero as denominator. Zero divided by a negative or positive number is either zero or is expressed as a fraction with zero as numerator and the finite quantity as denominator. Zero divided by zero is zero.

Really Brahmagupta is saying very little when he suggests that nn divided by zero is n/0n/0. Clearly he is struggling here. He is certainly wrong when he then claims that zero divided by zero is zero. However, it is a brilliant attempt from the first person that we know who tried to extend arithmetic to negative numbers and zero.

At this point, it must be noted here that there are some who use this to minimize our contribution to zero.

In A History of Mathematics, Chapter 10, Ancient and Medieval India, Pages 186-202, Carl B. Boyer, Uta C. Merzbach writes:

With the introduction, in the Hindu notation, of the tenth numeral, a round goose egg for zero, the modern system of numeration for integers was completed. Although the medieval Hindu forms of the ten numerals differ considerably from those in use today, the principles of the system were established. The new numeration, which we generally call the Hindu system, is merely a new combination of three basic principles, all of ancient origin: (1) a decimal base; (2) a positional notation; and (3) a ciphered form for each of the ten numerals. Not one of these three was originally devised by the Hindus, but it presumably is due to them that the three were first linked to form the modern system of numeration.

However, to repeat Joseph Needham’s statement, it must be taken as our then receptive:

But if China conserved with completeness her own characteristics style in mathematics until the beginning of the modern period, if India was the more receptive of the two cultures, one illustrious invention there was which seems to have occurred in the very border marches of the two great civilizations, into both of which it quickly spread. This was nothing less than development of a specific written symbol for nil value, emptiness, sunya, i.e. the zero. Perhaps we may venture to see in it an Indian garland thrown around the nothingness of the vacant space on the Han counting-board. (Source:  Science and Civilisation in China, Volume 3, Mathematics and the Sciences of the Heavens and the Earth, 19 Mathematics, Pages 148-149).

In conclusion, we can say that:

• While Babylonians may have had used wedge for zero, Greeks symbol for zero, and Chinese decimal place value system, it was Indians and only Indians who combined the various systems and made it a simple and very useful system as we have now.
• The full credit for evolving zero belongs to Indian intellectuals such as Aryabhata and Brahmagupta. However, there is hardly any role by vedas in this development.

SECTION E: MEDIEVAL TO MODERN TIMES: EVOLUTION OF TRIGONOMETRY IN INDIA AND DEVELOPMENT OF INFINITE SERIES OF TRIGONOMETRIC FUNCTIONS IN KERALA SCHOOL OF MATHEMATICS

Madhava of Sangagrama (AD 1340 to 1425) lived his life in Irinjalakuda, Thrissur, Kerala. His tremendous achievement was creating methods to compute accurate values for trigonometric functions by generating infinite series.

In the Crest of the Peacock, Non-European Roots of Mathematics, Chapter 10, A Passage to Infinity: The Kerala Episode, Pages 419-423, George Gheverghese Joseph provides the other key successors of Madhava of Sangagrama.

There are six texts that constitute the main evidence of the work of the Kerala school. They are Aryabhatiyabhasya (A Commentary on Aryabhatiya) and Tantrasamgraha (A Digest of Scientific Knowledge) of Nilakantha (1443–1544); Yuktibhasa (An Exposition of the Rationale) of Jyesthadeva (fl. 1500–1610); Kriyakramakari (Operational Techniques) of SankaraVariyar (c. 1500–1560) and Narayana (c. 1500–1575); Karanapaddhati (A Manual of Performances in the Right Sequence) of PutumanaSomayaji (fl. 1660–1740); and Sadratnamala (A Garland of Bright Gems) of Sankara Varman (1800–1838).

An important feature of these texts is their claim to have derived their principal ideas from Madhava of Sangamagrama (c. 1340–1425), who was mentioned earlier.

In the early 1820s, as J J O’Connor and E F Robertson writes in their article Charles Matthew Whish (https://mathshistory.st-andrews.ac.uk/Biographies/Whish/), three civil servants of East India Company assigned to under the local calendars and how those calendars were constructed. Those three were John Warren (1769-1830) and George Hyne, and Charles Matthew Whish (AD 1795 -1833). To quote J J O’Connor and E F Robertson:

The East India Company was interested in understanding local calendars and the methods used by the Hindus in constructing them. Warren seems to have been assigned such a task by the Company but the other two were also interested and became involved in the work. Whish discovered that certain Hindu texts contained approximations to π which had been found using series expansions. The question then arose as to whether the Hindus had developed these methods themselves or whether they had learnt them from Europeans.

Of the three assigned for the task, the youngest was Charles Matthew Whish. While John Warren and George Hyne dismissed the findings as Hindus learning infinite series from Europeans, Charles Matthew Whish finally took a stand that this was the original discovery of Hindus.

On July 1834, Charles Matthew Whish wrote an article in the Journal Transactions of the Royal Asiatic Society of Great Britain and Ireland / Volume 3 / Issue 03 titled “On the Hindú Quadrature of the Circle, and theinfinite Series of the proportion of the circumference to thediameter exhibited in the four S’ástras, the TantraSangraham, YuctiBháshá, CaranaPadhati, andSadratnamála”.

As you would notice, all the four sastras referred here by Charles Matthew Wish are those books of Kerala School of Mathematics. The author of Sadratnamala, Sankara Varman (1800–1838) was younger to Charles Matthew Wish as well.

Charles Matthew Whishwrote in his paper:

Having thus submitted to the inspection of the curious eight different infinite series, extracted from Brahmanical works for the quadrature of the circle, it will be proper to explain by what steps the Hindu mathematician has been led to these forms, which have only been made known to Europeans through the method of fluxions, the invention of the illustrious NEWTON. Let us first, however, know the age of these works; and as far as can be determined, the authors. First, then, it is a fact which I have ascertained beyond a doubt, that the invention of infinite series of these forms has originated in Malabar, and is not, even to this day, known to the eastward of the range of Ghats which divides that country, called in the earliest times Ceralam, from the countries of Madura, Coimbatore, Mysore, and those in succession, to that northward of these provinces.

While Charles Matthew Whish did not get enough attention at the time of publishing, it has gained attention since 1950s. At a popular level, one even would go the extend of claiming that Sir Isaac Newton stole the idea of Kerala School of Mathematics. At a scholarly level, there is a speculation that Kerala School of Mathematics could have influenced Newton and Others.

Kim Plofker aptly describes this. In the book, Mathematics in India Chapter 7, School of Madhava in Kerala, Page 217 to 253, Kim Plofker writes:

A similar struggle between speculation and prudence is provoked by the fascinating conceptual similarities between the Madhava school’s methods in infinite series and early modern European infinitesimal calculus techniques.

Some scholars have proposed that the former could have been the ultimate source of the latter. In broad outline, this hypothesis suggests that Jesuit missionaries in the vicinity of Cochin in the second half of the sixteenth century sought improved trigonometric and calendric methods in order to solve problems of navigation. Finding the Sine methods described in works of the Kerala school useful for that purpose, they transmitted this knowledge to their correspondents in Europe, whence it was disseminated via informal scholarly networks to the early developers of European calculus methods.

Although this idea is intriguing, it does not seem at present to have moved beyond speculation. There are no known records of sixteenth- or seventeenth-century Latin translations or summaries of these mathematical texts from Kerala. Nor do the innovators of infinitesimal concepts in European mathematics mention deriving any of them from Indian sources.

Is it a mere speculation without any evidence of transmission? Yes, George Gheverghese Joseph who is one of the proponents of this idea himself writes that it is a speculation at this point.

In the Crest of the Peacock, Non-European Roots of Mathematics, Chapter 10, A Passage to Infinity: The Kerala Episode, Page 438, George Gheverghese Joseph writes:

It is our conjecture that between 1560 and 1650 knowledge of Indian mathematical, astronomical, and calendrical techniques accumulated in Rome and diffused to neighboring Italian universities like Padua and Pisa, and to wider regions through Cavalieri and Galileo, and through visitors to Padua like James Gregory. Mersenne may have also had access to knowledge from India acquired by the Jesuits in Rome and, via his well-known correspondence, helped to diffuse this knowledge throughout Europe.

Certainly the way James Gregory acquired his geometry after his four-year sojourn in Padua, where Galileo taught, supports this possibility. All this is circumstantial. To make the case stronger for the transmission of Kerala mathematics to Europe, we require documentary evidence to show that the Jesuits acquired and comprehended mathematics of Kerala and disseminated this information among those in Europe who had the necessary background to assimilate this information. Failing the availability of such direct evidence, the indirect route for establishing transmission needs to be strongly delineated.

As George Gheverghese Joseph himself writes all this is just a conjecture even at this point, almost 200 years after Charles Matthew Whish first remarked about it.

In fact, in the case of Sir Isaac Newton, we can almost be certain that he was not aware of the Kerala School of Mathematics. We have his notebooks and those notes show no awareness of Kerala School of Mathematics. In A History of Mathematics, An Introduction, Chapter 8, Ancient and Medieval India, Page 260, Victor J Katz writes:

The most interesting question about transmission, however, relates to the power series for the sine and cosine. There is certainly no available documentation showing that any Europeans knew of the Indian developments in this area before the Europeans themselves worked out the power series in the mid-seventeenth century. However, there is some circumstantial evidence. First of all, Europeans, just like the Indians, needed precise trigonometric values for navigation. Secondly, the texts in which these power series were described were easily available in south India. Third, the Jesuits, in their quests to proselytize in Asia, established a center in south India in the late sixteenth century. In general, wherever the Jesuits went, they learned the local languages, collected and translated local texts, and then set up educational institutions to train disciples. But the question remains as to whether, in fact, the Jesuits did find these particular texts and bring them back in some form to Europe. As we will discuss in the chapters on calculus, in the period from 1630 to 1680 some of the basic ideas present in these Indian texts began to appear in European works. In the case of Newton, we can trace his thoughts through his notebooks and therefore have no reason to believe he was aware of Indian material. But for many of the other European mathematicians, we have little documentary evidence of how they discovered and elaborated on their ideas. So at the moment, we can only speculate as to whether Indian trigonometric series were transmitted in some form to Europe by the early seventeenth century.

It must be pointed out that there exists no hard evidence for transmission from Kerala School of Mathematics to even outside Kerala. In the Logic of Non-Western Science: Mathematical Discoveries in Medieval India, Daedalus, Fall, 2003, Vol. 132, No. 4, On Science (Fall, 2003), pp. 45-53, Published by: The MIT Press on behalf of American Academy of Arts & Sciences, David Pingree writes:

The discoveries of the successive generations of Madhava ‘school’ continued to be studied in Kerala within a small geographical area centred on Sangamagrama. The manuscript of the school’s Sanskrit and Malayalam treatises all copied in the Malayalam script, never travelled to another region of India; the furthest they got was Katattant in Kerala, about one hundred miles north of Sangamagrama, where RajakumaraSankra Varman repeated Madhava’s trigonometrical series in a work entitled Sadratanamala in 1823.

However, it is not merely the question of transmission – even more fundamental question is whether we can call trigonometric infinite series discovered by Madhava of Sangagrama as Calculus. Again, the answer is no.

In the Crest of the Peacock, Non-European Roots of Mathematics, Chapter 10, A Passage to Infinity: The Kerala Episode, Page 438, George Gheverghese Joseph writes:

Two powerful tools contributed to the creation of modern mathematics in the seventeenth century: the discovery of the general algorithms of calculus, and the development and application of infinite-series techniques. When introduced to calculus, one is often told that the names normally associated with the development of the subject are Newton and Leibniz. The other, less well-known stream, the discovery and applications of infinite series, is often downplayed despite its importance in the development of modern mathematics. Historically, the two streams tended to reinforce each other in their simultaneous development by each extending the range of application of the other.

While George Gheverghese Joseph showed the difference, in fact, it appears that he downplays the difference.

In Sherlock Holmes in Babylon and Other Tales of Mathematical History by Marlow Anderson, Victor Katz, Robin Wilson, Ideas of Calculus in Islam and India, Mathematics Magazine 68 (1995), 163-174, Victor J Katz writes:

How close did Islamic and Indian scholars come to inventing the calculus? Islamic scholars nearly developed a general formula for finding integrals of polynomials by A.D. 1000 | and evidently could find such a formula for any polynomial in which they were interested. But, it appears, they were not interested in any polynomial of degree higher than four, at least in any of the material which has so far come down to us. Indian scholars, on the other hand, were by 1600 able to use ibn al-Haytham’s sum formula for arbitrary integral powers in calculating power series for the functions in which they were interested. By the same time, they also knew how to calculate the differentials of these functions.

So some of the basic ideas of calculus were known in Egypt and India many centuries before Newton. It does not appear, however, that either Islamic or Indian mathematicians saw the necessity of connecting some of the disparate ideas that we include under the name calculus. There were apparently only specific cases in which these ideas were needed.

There is no danger, therefore, that we will have to rewrite the history texts to remove the statement that Newton and Leibniz invented the calculus. They were certainly the ones who were able to combine many differing ideas under the two unifying themes of the derivative and the integral, show the connection between them, and turn the calculus into the great problem-solving tool we have today. But what we do not know is whether the immediate predecessors of Newton and Leibniz, including in particular Fermat and Roberval, learned of some of the ideas of the Islamic or Indian mathematicians through sources of which we are not now aware.

In Sherlock Holmes in Babylon, Was Calculus Invented in India?, Mathematics Journal 33 (2002), 2-13, David Bressoud adds:

No. Calculus was not invented in India. But two hundred years before Newton or Leibniz, Indian astronomers came very close to creating what we would call calculus. Sometime before 1500, they had advanced to the point where they could apply ideas from both integral and differential calculus to derive the infinite series expansions of the sine, cosine, and arctangent functions:

This story provides illuminations of calculus that may have pedagogical implications. The traditional introduction of calculus is as a collection of algebraic techniques that solve essentially geometric problems: calculation of areas and construction of tangents. This was not the case in India. There, ideas of calculus were discovered as solutions to essentially algebraic problems: evaluating sums and interpolating tables of sines. Geometry was well developed in pre-1500 India. As we will see, it played a role. But it was, at best, a bit player. The story of calculus in India shows us how calculus can emerge in the absence of traditional geometric context. This story should also serve as a cautionary tale, for what did emerge was sterile. These mathematical discoveries led nowhere. Ultimately, they were forgotten, saved from oblivion only by modern scholars.

How do we resolve this? Was it part of calculus or not? David Pingree in his paper Hellenophilia versus the History of Science, makes an insightful observation. He remarks that when we study the history of science, we often tend to study through the lens of modern science. If there is any similarity between the modern science and the science of any other culture, we would tend to see similarity as part the system of modern science and not as an independent system.

David Pingreewrites in Hellenophilia versus the History of Science, Isis, Vol. 83, No. 4 (Dec., 1992), pp. 554-563 (10 pages):

I have already, if I have been at all successful, persuaded you that the fourth variety of Hellenophillia, in which one defines science as that which modern Western scientists believe in and the methodologies with which they operate, is inappropriate to a historian, though it may be useful to a modern Western scientists. And I have already mentioned that among the advantages provided to the historian by looking outside the confines of such restricted definition are a realization of potential diversity of interpretations of phenomena and of the actual diversity of the origins of ideas that have developed into modern Western science among other sciences, and an objectivity born of understanding of the cultural factors that impel sciences and scientists to follow one path rather than another. The loss of all these advantages is the price paid for suffering the passive effects of this form of Hellenophilia.

One example I can give you relates to the Indian Madhava’s demonstration, in about 1400 A.D., of the infinite power series of trigonometrical functions using geometrical and algebraic arguments. When this was first described in English by Charles Matthew Whish, in the 1830s, it was heralded as the Indians’ discovery of the calculus. This claim and Madhava’s achievements were ignored by Western historians, presumably at first because they could not admit that an Indian discovered the calculus, but later because no one read anymore the Transactions of the Royal Asiatic Society, in which Whish’s article was published. The matter resurfaced in the 1950s, and now we havethe Sanskrit texts properly edited, and we understand the clever way that Madhavaderived the series without the calculus; but many historians still find it impossible toconceive of the problem and its solution in terms of anything other than the calculusand proclaim that the calculus is what Madhava found. In this case the elegance andbrilliance of Madhava’s mathematics are being distorted as they are buried underthe current mathematical solution to a problem to which he discovered an alternateand powerful solution.

To highlight the two important sentences in this paragraph:

• And I have already mentioned that among the advantages provided to the historian by looking outside the confines of such restricted definition are a realization of potential diversity of interpretations of phenomena and of the actual diversity of the origins of ideas that have developed into modern Western science among other sciences, and an objectivity born of understanding of the cultural factors that impel sciences and scientists to follow one path rather than another.
• The matter resurfaced in the 1950s, and now we have the Sanskrit texts properly edited, and we understand the clever way that Madhava derived the series without the calculus; but many historians still find it impossible to conceive of the problem and its solution in terms of anything other than the calculus and proclaim that the calculus is what Madhava found

In other words, Madhava derived infinite series WITHOUT calculus  and there were cultural factors that impelled him to follow that path compared to the path taken by Newton.

Before we summarize this section, let us highlight one positive aspect of this discussion. With the developments of Kerala School of Mathematics brought to light, we now know that trigonometry in India had a continuous development. While Trigonometry had its birth in Greece, it spent its childhood and youth in India.

In Sherlock Holmes in Babylon, Was Calculus Invented in India?, Mathematics Journal 33 (2002), 2-13, David Bressoud writes about the birth of trigonometry in Greece:

Trigonometry arose from, and for over fifteen hundred years was used exclusively for, the study of astronomy/astrology. Hipparchus of Nicaea (ca. 161-126 BC) is considered the greatest astronomer of antiquity and the originator of trigonometry. Trigonometry was born in response to a scientific crisis. The Greek attempt to cast astronomy in the language of geometry was running up against the disturbing fact that the heavens are lop-sided. New tools were needed for analyzing astronomical phenomena.

Let me paint the background to this crisis. It begins with the assumption that the earth is stationary. While this was debated in early Greek science does the earth go around the sun or the sun around the earth?- the simple fact that we perceive no sense of motion is a powerful indication that the earth does not move. In fact, when in the early seventeenth century it became clear that the earth revolves about the sun, it created a tremendous problem for scientists: How to explain how this was possible? How could it be that we were spinning at thousands of miles per hour and hurtling through space at even greater speeds without experiencing any of this? Surely if the earth did move, we would have been flung off long ago. Newton’s great accomplishment in the Principia was to solve this problem. He created inertial mechanics for this purpose, building it with the then nascent tools of calculus. So we begin with a fixed and immovable earth.

George Ghevarghese Joseph summarizes the development of trigonometry in India till Madhava of Sangagrama. In The Crest of the Peacock, Non-European Roots of Mathematics, Chapter 9, Page 394-395, he writes:

While the work of the Alexandrians Hipparchus (c. 150 BC), Menelaus (c. AD 100), and Ptolemy (c. AD 150) in astronomy laid the foundations of trigonometry, further progress was piecemeal and spasmodic. From about the time of Aryabhata I (c. AD 500), the character of the subject changed, and it began to resemble its modern form. Subsequently it was transmitted to the Arabs, who introduced further refinements. From the Islamic world, the knowledge spread to Europe, where a detailed account of existing trigonometric knowledge first appeared under the title De triangulis omni modis, written in 1464 by Regiomontanus. In early Indian mathematics, trigonometry formed an integral part of astronomy. References to trigonometric concepts are found in the Surya  siddhanta (c. AD 400), Varahamihira’sPancasiddhantika (c. AD 500), and Brahmagupta’s Brahma Sphuta-¬siddhanta (AD 628). A detailed and systematic study of the subject was made by Vatesvara (b. AD 880) in the VatesvaraSiddhanta and then by Bhaskaracharya in his SiddhantaSiromani. He felt that the title acharya (i.e., master or teacher) in astronomy could be given only to those who possessed sufficient knowledge of trigonometry. Infinite expansions of trigonometric functions, building on Bhaskaracharya’s work, are found in the work of Madhava and Nilakantha, discussed chapter 10.

Now to summarize the major points of discussion:

• Trigonometry which began in Greece developed in India and it continued to develop even after the Golden Age of Mathematics in India as we see it from Kerala School of Mathematics
• The infinite trigonometric series developed in Kerala was without calculus. It was a different way in a different cultural context
• There is no evidence for transmission of this knowledge to the Western countries before Westerners discovered it themselves. In fact, in Sir Isaac Newton’s case, we can be certain that he had no awareness of Kerala School of Mathematics

Conclusion:

We began our discussion with the question whether the mathematical contributions of India are derived from Vedas or whether it is like any other human endeavour of receiving from others and giving to others. In the case of the book of Vedic Mathematics by Swami Bharati Krishna Tirtha, we have seen that sutras mentioned by Swami have no reference whatsoever in any Sanskrit literature. In the case of Pythagoras theorem, we can rightly claim to have recorded it first as a theorem though Egyptians and Babylonians were aware of it before us. In other cases, whether it is the contribution of zero or development of trigonometry, it was both a give and take- a sister civilization.

At this point, we must ask the question – if we had brilliant men such as Madhava of Sangagrama, why didn’t they contribute like Sir Isaac Newton. In Sherlock Holmes in Babylon, Was Calculus Invented in India?,Mathematics Journal 33 (2002), 2-13, David Bressoudlaments:

There is no evidence that the Indian work on series was known beyond India, or even outside Kerala, until the nineteenth century. Gold and Pingree [4] assert that by the time these series were rediscovered in Europe, they had, for all practical purposes, been lost to India. The expansions of the sine, cosine, and arc tangent had been passed down through several generations of disciples, but they remained sterile observations for which no one could find much use. No. Calculus was not discovered in India. I am left wondering how much important mathematics is today known but not yet discovered, passed among a coterie of tightly knit disciples as an intriguing yet seemingly useless insight, lacking the context, the fertilizing connections, that would enable it to blossom and produce its fruit.

If Madhava of Snagagrama could not contribute as Sir Isaac Newton, it was of course not the lack of talent or brilliance but the lack of a proper culture. Sir Isaac Newton had a proper Biblical cultural context which condemned astrology but promoted astronomy and “laws of nature”, whereas Madhava of Snagagrama lived in a culture that promoted astrology and occultic character of nature. Madhava of Snagagrama discovered what was required of him by his culture and Sir Isaac Newton discovered what was required of him by his culture. Vedic followers instead of taking credit must take the blame of not providing a proper culture for several madhavas to become Newton and Einstein.

 

APPENDIX 1 – SWAMI BHARTI KRISHNA TIRTHA PREFACE TO VEDIC MATHEMATICS

 

AUTHOR’ S PREFACE

A —A DESCRIPTIVE PREFATORY NOTE ON THE ASTOUNDING WONDERS OF ANCIENT INDIAN VEDIC MATHEMATICS

1. In the course of our discourses on manifold and multifarious subjects (spiritual, metaphysical, philosophical, psychic, psychological, ethical, educational, scientific, mathematical, historical, political, economic, social etc., etc., 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 Indian scriptures, nay, the oldest “ Religious” scriptures of the whole world) claim to deal with all branches 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.
2. The very word “Veda” has this derivational meaning i.e. the fountain-head and illimitable store-house 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 other-worldly) matters but also to those usually described as purely “secular”, “temporal”, or “wordly”; 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.
3. 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 matters which any aspiring seeker after knowledge can possibly seek to be enlightened on.
4. It is thus in the fitness of things that the Vedas include (i) Ayurveda (anatomy, physiology, hygiene, sanitary science, medical science, surgery etc., etc.,) not for the purpose of achieving perfect health and strength in the after-death future but in order to attain them here and now in our present physical bodies; (ii) Dhanurveda (archery and other military sciences) not for fighting with one another after our transportation to heaven but in order to quell and subdue all invaders from abroad and all insurgents from within; (iii) Gandharva Veda (the science and art of music) and (iv) Sthapatya Veda (engineering, architecture etc and all branches of mathematics in general). All these subjects, be it noted, are inherent parts of the Vedas i.e. are reckoned as “ spiritual” studies and catered for as such therein.
5. Similar is the case with regard to the Vedangas (i.e. grammar, prosody, astronomy, lexicography etc., etc.,) which, according to the Indian cultural conceptions, are also inherent parts and subjects of Vedic (i.e. Religious) study.
6. As a direct and unshirkable consequence of this analytical and grammatical study of the real connotation and full implications of the word “ Veda” and owing to various other historical causes of a personal character (into details of which we need not now enter), we have been from our very early childhood, most earnestly and actively striving to study the Vedas critically from this stand-point and to realise and prove to ourselves (and to others) the correctness (or otherwise) of the derivative meaning in question.
7. There were, too, certain personal historical reasons why in our quest for the discovering of all learning in all its departments, branches, sub-branches etc., in the Vedas, our gaze was riveted mainly on ethics, psychology and metaphysics on the one hand and on the “positive” sciences and especially mathematics on the other

 

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

 

9. And we were agreeably astonished and intensely gratified to find that exceedingly tough mathematical problems (which the mathematically most advanced present day Western scientific world had spent huge lots of time, energy and money on and which even now it solves with the utmost difficulty and after vast labour involving large numbers of difficult, tedious and cumbeisome “steps” of working) can be easily and readily solved with the help of these ultra-easy Vedic Sutras (or mathematical aphorisms) contained in the Parisista (the Appendix portion) of the ATHARVAVEDA in a few simple steps and by methods which can be conscientiously described as mere “mental arithmetic”.
10. Ever since (i.e. since several decades ago), we have been carrying on an incessant and strenuous campaign for the India-wide diffusion of all this scientific knowledge, by means of lectures, blackboard- demonstrations, regular classes and so on in schools, colleges, universities etc., all over the country and have been astounding our audiences everywhere with the wonders and marvels not to say, miracles of Indian Vedic mathematics.

 

11. We were thus at last enabled to succeed in attracting the more than passing attention of the authorities of several Indian universities to this subject. And, in 1952, the Nagpur University not merely had a few lectures and blackboard-demonstrations

given but also arranged for our holding regular classes in Vedic mathematics (in the University’s Convocation Hall) for the benefit of all in general and especially of the University and college professors of mathematics, physics etc.

 

12. And, consequently, the educationists and the cream of the English educated section of the people including the highest officials (e.g. the high-court judges, the ministers etc.,) and the general public as such were all highly impressed ; nay, thrilled, wonder-struck and flabbergasted ! And not only the newspapers but even the University’s official reports described the tremendous sensation caused thereby in superlatively eulogistic terms; and the papers began to refer to us as “ the Octogenarian Jagadguru Shankaracharya who had taken Nagpur by storm with his Vedic mathematics” , and so on !

 

13. It is manifestly impossible, in the course of a short note (in the nature of a “ trailer” ), to give a full, detailed, thorough- going, comprehensive and exhaustive description of the unique features and startling characteristics of all the mathematical lore in question. This can and will be done in the subsequent volumes of this series (dealing seriatim and in extenso with all the various portions of all the various branches of mathematics).

 

14. We may, however, at this point, draw the earnest attention of every one concerned to the following salient items thereof:—

 

(i) 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. In fact, there is no part of mathematics, pure or applied, which is beyond their jurisdiction

 

(ii) The Sutras are easy to understand, easy to apply and easy to remember ; and the whole work can be truthfully summarised in one word “ mental” !

 

(iii) Even as regards complex problems involving a good number of mathematical operations (consecutively or even simultaneously to be performed), the time taken by the Vedic method will be a third, a fourth, a tenth or even a much smaller fraction of the time required according to modern (i.e. current) Western methods;

 

(iv) And, in some very important and striking cases, sums requiring 30, 50, 100 or even more numerous and cumbrous “ steps” of working (according to the current Western methods) can be answered in a single and simple step of work by the Vedic method ! And little children (of only 10 or 12 years of age) merely look at the sums written on the blackboard (on the platform) and immediately shout out and dictate the answers from the body of the convocation hall (or other venue of the demonstration). And this is because, as a matter of fact, each digit automatically yields its predecessor and its successor ! and the children have merely to go on tossing off (or reeling off) the digits one after another (forwards or backwards) by mere mental arithmetic (without needing pen or pencil, paper or slate etc) !

 

(v) On seeing this kind of work actually being performed by the little children, the doctors, professors and other “ big-guns” of mathematics are wonder struck and exclaim :—“ Is this mathematics or magic” ? And we invariably answer and say : “It is both. It is magic until you understand it ; and it is mathematics thereafter” ; and then we proceed to substantiate and prove the correctness of this reply of ours ! And

 

(vi) As regards the time required by the students for mastering the whole course of Vedic mathematics as applied to all its branches, we need merely state from our actual experience that 8 months (or 12 months) at an average rate of 2 or 3 hours per day should suffice for completing the whole course of mathematical studies on these Vedic lines instead of 15 or 20 years required according to the existing systems of the Indian and also of foreign universities

 

15. In this connection, it is a gratifying fact that unlike some so-called Indologists (of the type hereinabove referred to) there have been some great modern mathematicians and historians of mathematics (like Prof. G. P. Halstead, Professor Ginsburg, Prof. De Moregan, Prof. Hutton etc.,) who have, as truth-seekers and truth-lovers, evinced a truly scientific attitude and frankly expressed their intense and whole-hearted appreciation of ancient India’s grand and glorious contributions to the progress of mathematical knowledge (in the Western hemisphere and elsewhere).

 

16. The following few excerpts from the published writings of some universally acknowledged authorities in the domain of the history of mathematics, will speak eloquently for themselves:—

 

(i) On page 20 of his book “ On the Foundation and Technique of Arithmetic” , we find Prof. G.P. Halstead saying “ The importance of the creation of the zero mark can never be exaggerated. This giving of airy nothing not merely a local habitation and a name, a picture but helpful power is the characteristic of the Hindu race whence it sprang. It is like coining the Nirvana into dynamos. No single mathematical creation has been more potent for the general on-go of intelligence and power”.

 

(ii) In this connection, in his splendid treatise on “ The present mode of expressing numbers” (the Indian Historical Quarterly Vol. 3, pages 530-540) B. B. Dutta says: “ The Hindus adopted the decimal scale vary early. The numerical language of no

other nation is so scientific and has attained as high a state of perfection as that of the ancient Hindus.

 

In symbolism they succeeded with ten signs to express any number most elegantly and simply. It is this beauty of the Hindu numerical notation which attracted the attention of all the civilised peoples of the world and charmed them to adopt it”

 

(iii) In this very context, Prof. Ginsburg says:— “ The Hindu notation was carried to Arabia about 770 A.D. by a Hindu scholar named Kanka who was invited from Ujjain to the famous Court of Baghdad by the Abbaside Khalif Al-MANSUR. Kanka taught Hindu astronomy and mathematics to the Arabian scholars ; and, with his help, they translated into Arabic the Brahma-Sphuta-Siddhanta of Brahma Gupta. The recent discovery by the French savant M.F. Nau proves that the Hindu numerals were well

known and much appreciated in Syria about the middle of the 7th Century A -D ” . (G insburg’s“ New Light on our numerals” , Bulletin of the American Mathematical Society, Second series, Vol. 25, pages 366-369).

 

(iv) On this point, we find B. B. Dutta further saying : “ From Arabia, the numerals slowly marched towards the West through Egypt and Northern Arabia; and they finally entered Europe in the 11th Century. The Europeans called them the Arabic notations, because they received them from the Arabs. But the Arabs themselves, the Eastern as well as the Western, have unanimously called them the Hindu figures. (Al-Arqan-Al-Hindu”.)

17. The above-cited passages are, however, in connection with and in appreciation of India’s invention of the “ Zero” mark and her contributions of the 7th century A.D. and later to world mathematical knowledge.

In the light, however, of the hereinabove given detailed description of the unique merits and characteristic excellences of the still earlier Vedic Sutras dealt with in the 16 volumes of this series, the conscientious (truth-loving and truth-telling) historians of Mathematics (of the lofty eminence of Prof. De Morgan etc.) have not been guilty of even the least exaggeration in their candid admission that “ even the highest and farthest reaches of modern Western mathematics have not yet brought the Western world even to the threshold of Ancient Indian Vedic Mathematics” .

18. It is our earnest aim and aspiration, in these 16 volumes*, to explain and expound the contents of the Vedic mathematical Sutras and bring them within the easy intellectual reach of every seeker after mathematical knowledge.

*Only one volume has been bequeathed by His Holiness to posterity cf p. x above—General Editor.