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Research Into Indic Mathematicians
I proposed elsewhere that data for Groups (powers) of Tens or Dashatis in the Rgveda was expressed with two words: Pad and Hasta. Both words denote the limbs of human beings, each containing its group of tens, counted as the total of toes respectively fingers.

There is a Rc which exactly formulates what I've proposed. See below, under Niyuta in RV.

Previously, I thought that the highest number with a name in the Rgveda was Ayuta=10.000. This is true for the older layers off the Rgveda. I have found even higher named numbers, which belong to a later layer. See below, under Niyuta and ParArdha in RV.


The searching started with this counting unit. Rc IV.58.3 of VAmadeva Gautama gives the sequence 432 and 7 Hastas or 10.000.000. I proposed a Vaidika counting along this line: 1 Hasta = 1x10, 2 hastas = 1 Hasta x 1 Hasta or 10x10=100, 3 Hastas = 2 Hasta x 1 Hasta or 100x10=1000, etc.

This Rc suggests the huge number 4.320.000.000 or a number identified later as a Kalpa.


Then I advanced with another Rc containing a concurring counting unit:

– Rc I.164.48 in the SUkta of RSHi DirghatamA Aucathya gives huge numbers starting with Ekapad or 10 to Navapad or in Parame Vyoman or in the Highest Space. Here we see that GaurI has lowed or a shining object made noie and formed or generated multifold Salilas or waterfloods. A Gaura is the Horse-like speedy onager (Ger in ona-ger and Gur in Persian are related to the word Gaura which has its local form Khur in the Rann of Kutch – see the paper of prof. R.N. Iyengar). The female alien GaurI was joined with the (land of the) local Gaura.

Additional to this Rc, I found the next one:

– Rc X.117.8 of RSHi BhikSHu AngirA gives the ratio 4:3:2:1. The numbers are in Pads preceded with the sacred sequence 432. Thus, we have 432 x 10-100-1000-10.000 or 4320-43.200-432.000-4.320.000.

Huge numbers

The Yajurveda gives the groups of tens sequence from 10 to 1.000 billion or ParArdha. (While Shukla Yajurveda gives the sequence ayuta-niyuta-prayuta, KYV KS has the older sequence ayuta-prayuta-niyuta. It seems that the PaippalAda Atharvaveda supports the last with asking for Ayuta or 10.000 to become Prayuta or 100.000 (ayutam prayutam bhava).

Niyuta in RV

Now, I have found a new number in the Rgveda, that not only is higher than Ayuta, but even clearly proofs that higher number with the Pad counting, as I had proposed:

nákSHad dhávam aruNiíH puurvyáM raáT turó vishaám áÑgirasaam ánu dyuún |

tákSHad vájraM níyutaM tastámbhad dyaáM cátuSHpade náryaaya dvipaáde ||

Rc I.121.3 of RSHi KakSHIvAn Dairghatamasa, says that Dyo was fixed, forming a Niyuta-fold Vajra for the DvipAda CatuSHpad Narya. Here we have a juxtapositioning of niyuta+vajra versus dvipAda-catuShpad+narya, where the vajra has a connection with narya. As we are dealing here with counting and numbers, Niyuta must be a huge number equal to a DvipAda-CatuSHpad.

DvipAda as 10x10=100 and catuSHpad as 1000x10=10.000 gives 100 Ayuta or 1.000.000. And this is called Niyuta in Rk-pAda c.

Getting support of AV PaippalAda and KYV-KATHaka we know that after Ayuta or 10.000 comes Prayuta or 100.000. And KATHaka continues then with Niyuta or 1.000.000 (this is also the case in later literature, Br., Pur. As per MW dictionary). Thus, RV Niyuta = dvipAda-catuSHpad = 1.000.000!

ParArdha in RV

The Asya VAmasya SUkta provides us with even huger numbers than the Niyuta. In I.164.12 the RSHi of that Rc gives other units to provide us with huge numbers.

páñcapaadam pitáraM dvaádashaakRtiM | divá aahuH páre árdhe puriiSHíNam |

áthemé anyá úpare vicakSHaNáM | saptácakre SHáLara aahur árpitam ||

They speak of the PancapAda Father as DvAdasha-AkRti in the upper region of heaven reaching the marshy shores.

PancapAda = 100.000 and DvAdasha-AkRti =, which is exactly a ParArdha of the Yajurvedas. Here we get explained the name of the huge number.

The second Rgardha explains the gap below (upare) between PancapAda and Dvadasha-AkRti as Sapta-cakra, starting with SHaL-ara or 1.000.000.

Remember that Rc I.164.48 first mentions the lower Pads, then leaves the first region of Vyoman, containing

5-6-7 Pads, to reach Parame Vyoman with 8-9 Pads. The number for Dashapad is Madhya in Yajurveda and for EkAdashapad is Anta. The numbername Madhya or intermediate, is between a Samudra (Navapad) and a next region, which must be containing Anta and ParArdha. As the last is Para-Ardha, the previous is the ordinary Ardha. Thus, after Samudra comes Madhya and then Ardha= YV Anta and finally Para-Ardha.

Note: What amazes me, is that the RSHi starts with the many saline Salilas coming towards a single Samudra and then reaches ParArdha in the PurISHin shores. This means that the RSHi anticipates having knowledge of or supposes new regions beyond ParArdha in the remotest sky (Div).

We can safely conclude:

a. that the Rgveda counting was a decimal one, with names for each number. gave Eka to Dasha upto ParArdha or 1-10 to at least identified till now.

b. the RSHis of that work also used a different counting, in groups or powers of tens, with the twofold unit names of Pad (var. PAda) and Hasta.

c. they also were very familiar with the 4:3:2:1 ratio – huge numbers upto a named Niyuta and Pad or Hasta counting were multiplied with a preceding sacred number 432.

I haven't yet touched the device of diverse ChandA syllables for counting purposes.

Caturyuga division

These huge numbers are also part of the later PaurANika and SaiddhAntika systems. The PurANas are evolved out of the Vaidika PuRANa-veda and the combined Vaidika ItihAsa-PurANa. The ratio 4:3:2:1 coupled to these huge numbers, combined with the Vaidika AkSHa game with VibhItakas (see Xth MaNDala) containing that ratio led to the Caturyuga system.

The RSHis were also fond of DvAdashatis or duodecimal counting: 12-24-36-48-60-72-84-96-108-120-.... 360, ….. 432, etc. it is particularly this counting which led them to their sacred numbers, and secondarily to their BArhaspatya or sexagesimal counting.

The Rgvaidika 432 sapta-hasta number is 360x12.000 or a Kalpa. The Rgvaidika 432 PancapAda (Ayuta) is exactly a MahAyuga.

For a fractional division of the Pad in four, see I.164.45 (about VAk):

A. catvaári vaák párimitaa padaáni taáni vidur braahmaNaá yé maniiSHíNaH |

a. catvaári vaák párimitaa padaáni | b. taáni viduH braahmaNaáH yé maniiSHíNaH |

B. gúhaa triíNi níhitaa néÑgayanti turiíyaM vaacó manuSHyaaaà vadanti ||

c. gúhaa triíNi níhitaa ná iÑgayanti | d. turiíyam vaacáH manuSHyaàH vadanti ||

Speech hath been measured out in four divisions, the Brahmans who have understanding know them.

Three kept in close concealment cause no motion; of speech, men speak only the fourth division.

Here we again have a 4:3:2:1 = 10 ratio within a Pad = 10. An additional factor is that the 4/4th, 3/4th and 2/4th Pad parts are placed versus the 1/4th part. Just like, analogous to this, in Caturyuga counting the 1/4th Kali-Yuga is placed versus the other 3/4th Yugas in a Dharma sense. The AkSHa game and counting according to the 4:3:2:1 ratio came together. Remember that AkSHa is not only a dice game using the ratio 4:3:2:1, but it is by its name also connected with Time cycli: the Wheel of Time of DIrghatamA with 360 spokes has a nave or nAbhi consisting of three parts: 1. the ring, the Kha hole and the AkSHa pole connecting this 360 days wheel with higher wheels or period cycles. (Perhaps this threefold nAbhi, especially the Kha hole is the origin of the Indian zero sign. You first have the the nAbhi, then come the numbers from 1-360. The three NAbhis also point to 360/3=120 or Three CAturmAsikAs.)

1. Using the 4:3:2:1 ratio with the Rgvaidika MahAyuga number gives the duration of each single Yuga => 1.728.00 : 1.296.000 : 864.000 : 432.000. The last is a 432 Tripad.

2. Using the ratio 4:3:2:1 with the number 12.000 gives the duration of a Caturyuga => 4800-3600-2400-1200. (4:3:2:1 ratio in DvAdashatis gives 48:36:24:12)

The number 12, thus, was very special to the Vaidika people, which explains their Twelve-Year Sattra. DIrghatamA gives in his Asya VAmasya SUkta the image 360 spokes and 12 fellies. This 360 is one year multiplied by 12 = 4320 days in 12 years.

The 4320 = 360x12 years. But it also comes to 365x11,8 or 366x11,8 minus approximately a CAturmAsikA.

Vaidika DvA-dashatis

The ancient RSHis arrived at different sacred number, like 60, 72, 108, 360, 432 and their multiplication with the decimal Pad or Hasta units, because they also counted in 12s, a duodecimal system. Combining the DvA-dashati numbers within a Dashati table gave this:

000-001-002-003-004—005—006-007-008-009 (Dashati table)

DvAdashati series in a Dashati counting





In this DvA-dashati we have all the basis ingredients for a CaturyugI system (the number 432=36x12, and further at positions 004-003-002-001 we have the numbers 48-36-24-12, there is a 72, 108), but also 360 for days, 12 for months of 30 days.

(Perhaps this was put on a Vaidika IriNa-board, evolving in a ten x ten Pad or PAda (Dashapad or -pAda board. This must have facilitated counting. In later times we have the Chinese and Indians using the board for counting, see Plofker)

A last word, on the Aucathyas

The Aucathyas composed SUktas in the Rgveda which amounts to 40, almost a Family MaNDala. I call these the Aucathya CatvAriMsha RkKANDa. A Rk-KANDa of the Aucathya Kula, consisting of 25 SUktas of DIrghatamA Aucathya, 12 of KakSHIvAn Dairghatamasa (11 in the Ist and 1 in the IXth MaNDala) and 3 of Ucathya Angirasa (in the IXth).

About SUkta I.164, the huge numbers are those composed in JagatI ChandA. The number Niyuta is of KakSHivAn, which means he is earlier in time. Therefore, I conclude that the TriSHTubh Rcs in I.164 are those of DIrghatamA, and the JagatI ones are those of an Aucathya descendant, later in time than KakSHIvAn. And the latest Aucathya descendant, at a time just before the inclusion of the Ist MaNDala, must have finished the extended AprI SUkta in AnuSHTubh.

RSHi DIrghatamA composed the Rcs in TriSHTubh containing amongst others four important ones containing DvA suparNA (20), Ekam sadvipra (46), and PrchAmi tvA param antam (34) and DvAdasha pradhayash cakram (48).

The decendants of RSHi DIrghatamA elaborated further on his concept, leading to measuring the limits of the sky (caused by Rc 164.34) towards ParArdha in Div or furthest sky giving the figure of 1.000 billion.

BhikSHU AngirA also used the Pad unit counting, whereas VAmadeva Gautama of another AngirA branch, distantly related to the Aucathyas, used a Hasta unit.
I should do a complete review of Kim Plofkers book . It is not a bad book, and she does use the adjective brilliant towards the end of the book. but i sill have reservations about the manner in which she approaches the subject.I will give an excerpt from my book,which is relatively mild in its criticism of Plofkers ( in contrast to her guru David Pingree, who does not fare well in my book)

Plofker , p. 293 Quote: "It is not unusual to encounter in discussions of Indian mathematics such assertions as that “the concept of differentiation was understood [in India] from the time of Manjula (... in the 10th century)” [Joseph 1991, 300], or that “we may consider Mādhava to have been the founder of mathematical analysis” (Joseph 1991, 293), or that Bhāskara II may claim to be “the precursor of Newton and Leibniz in the discovery of the principle of the differential calculus” (Bag 1979, 294). ... The points of resemblance, particularly between early European calculus and the Keralese work on power series, have even inspired suggestions of a possible transmission of mathematical ideas from the Malabar coast in or after the 15th century to the Latin scholarly world (e.g., in (Bag 1979, 285)). ... It should be borne in mind, however, that such an emphasis on the similarity of Sanskrit (or Malayalam) and Latin mathematics risks diminishing our ability fully to see and comprehend the former. To speak of the Indian “discovery of the principle of the differential calculus” somewhat obscures the fact that Indian techniques for expressing changes in the Sine by means of the Cosine or vice versa, as in the examples we have seen, remained within that specific trigonometric context. The differential “principle” was not generalized to arbitrary functions—in fact, the explicit notion of an arbitrary function, not to mention that of its derivative or an algorithm for taking the derivative, is irrelevant here".

But Plofker’s objections to assigning credit to the Indians cannot be taken seriously , since most science is developed by observing specific instances and then in the course of time it is possible to see a pattern which permits the generalization to a broader principle , so when she says that application of a derivative to a trigonometric functions does not make a calculus, she is being disingenuous, because Indic mathematicians were already applying analysis to a large class of functions. It must be conceded that Europe steadily drew ahead after 1700, but the reasons for that have little to do with the work that Indic mathematicians did in the past.

I am appending excerpts from the Preface to the book

This is the title of the book which is Volume 5 of the Series on Distortions in Indian History. The book is being released with immediate effect for public consumption. The book deals with the effects of Colonialism and Eurocentricity. Colonialism is not merely the conquest of dominion of vast lands and exerting your will on millions of people. It is more than the act of unleashing violence, a violence not restricted to the physical realm. It subjects the colonized to an epistemic rupture of vast proportions. This is the narrative of one example of such an epistemic rupture. We tell the story ( and the history ) of such a rupture in the case of Astronomy and Mathematics in the Indian suibcontiennt

(PRWEB) September 12, 2010 -- This book is the narrative of one example of an epistemic rupture, of the theft and destruction of i the cultural heritage. We tell the story ( and the history ) of such a rupture in the case of Astronomy and Mathematics. Indic Studies Foundation is pleased to announce the release of Volume 5 of the Series on Distortions in Indian History. The book is available at http://www.lulu.com/product/hardcover/th...f/center/4

The main reason for writing this book is that the real story of the Indic contribution to Astronomy has yet to be told in the text books of the west. Few books(1,2) give a coherent account of the Indic odyssey as it unfolds from the mists of antiquity to the pioneering work of Astrophysicist Chandrasekhar on the nature of the universe. If they do mention it at all, it is merely to say that they borrowed everything from Greece or Babylon. When challenged, the mathematician in the west will quote one of the 3 or 4 Occidental historians of Mathematics (e.g. Toomer, Van der Waerden or David Pingree) as their authoritative source. Rarely will they mention a Primary source in Sanskrit, because they are not familiar with the literature in Sanskrit and they do not trust the Indics to tell the true story. They prefer to get the story from an Occidental who may not have read a single book in its Sanskrit original rather than get it from an Indic. The net result is a book filled with clichés where the content is already degraded from multiple levels of interpretation and inadvertent filtering of the original source.

Part of the reason for writing this book is to influence all the readers, regardless of their ethnicity, ideology, or geography to adopt a more global perspective on matters relating to History and philosophy of the sciences. Under such a perspective, few would feel compelled to defend or attack a viewpoint if the extent of the antiquity was the sole issue at stake. But the yearning for a competitive antiquity is not restricted to those of a particular ethnicity. It appears to be a predominant factor when a more aggressive and authoritarian civilization subjugates a people with a more advanced episteme. Time and again , this pattern of behavior has been the norm, where the aggressor has adopted the Epistemes of the subjugated people, after devoting a massive effort to absorb the knowledge, and once he is fairly confident that he has been successful in this endeavor, he will turn around and assert precisely the opposite, that in fact it is the subjugated civilization that has borrowed the episteme and the resulting knowledge.

Keep in mind that antiquity affects many factors that have a bearing on the sense of uniqueness that a people have of themselves and a sense that continuity and longevity of a civilization bestows a modicum of a sense of wellbeing. A loss of epistemic continuity that is now being experienced in the Indian subcontinent has long term consequences for the manner in which the Indics will look upon themselves. Civilization is a fragile thing, if I may paraphrase Will Durant, the great historian who compiled the monumental Story of Civilization over a thirty year period, and it does not take much to obliterate a civilization.

The Indic approach to astronomy, contrary to presuppositions in the occident was characterized by mundane motivations namely the need to determine accurately the date, time and place of the location of the main planetary bodies that he could see with the naked eye; the Sun, Moon, Venus, Mercury, Jupiter and Saturn in relation to the Earth. As we shall repeat on more than one occasion, in this endeavor he was eminently practical. Again, his motivation for the determination of these quantities was also driven by pragmatic considerations like a need for fixing the seasons when planting was necessary. He may have found it politic to cloak these mundane considerations in ritualistic garb in order to impress those in society not blessed with the analytical skills that he may have used. In summary the goal of the Ancient Indic Prayojana or raison-de-être of the stra was primarily the determination of Kalá (Time), Dik (direction or orientation), and Desha (Place)

The story of the calendar and the development of mathematics and astronomy is indeed a fascinating chapter in the intellectual history of the species. It is laced with people of superior talents, but all too often these very same gifted individuals were not able to rise above petty considerations, while they were uncovering the secrets of the skies.

It is unfortunate that the Indic role in this fascinating chapter has been largely ignored in most western descriptions of the history of astronomy and time. There hardly exists a history book in Astronomy that does justice to the fact that the ancient Indic left behind a staggering amount of literature for us to decipher. In fact the perception is just the opposite that Information about Indian math is hard to get. This is in large part a problem that the occidental has created by imposing very high standards of reliability, in many cases standards that are impossible to meet and certainly were never demanded of similar sources from Ancient Greece. As a result the bias against Indic contributions in antiquity has been institutionalized to a large degree. This is regrettable and as a result, the story within a story of how the occidental tried to ignore, minimize and even suppress the Indic contribution is equally interesting.

This book is not about the glories of a bygone era, where one bemoans the ephemeral nature of an enlightened past. It is a recounting of the irreversible nature of the changes that take place when a civilization is subjugated. Its traditions are ridiculed. Its history is rewritten, its language is driven into oblivion and any attempt to combat this assault albeit in a non-violent and scholarly manner, marks the individual as a fundamentalist. The calendar, astronomy, and the story of time combine to make a fascinating chapter in the story of the homo sapien, but it is to be hoped tha it is the larger Civilizational canvas that the reader will focus on.

What do I take away from the writing of this book? My faith in the universality of the human spirit. If there is one thing above all that I treasure from this experience is that the love of science and mathematics does not recognize man made geographies, boundaries, ethnic classifications, language, social strata or economics. It is for this reason I find that the current Eurocentrric emphasis in the Occident which persists among authors even to this day to be a anathema and to be of a particularly egregious nature with which I have little sympathy and have no tolerance whatsoever. The book can be ordered from www.lulu.com. References

1.See for instance, James Evans, The history and practice of Ancient Astronomy, Oxford University press, New York,1998 2.Hoskin, Michael, "The Cambridge Concise History of Astronomy”, Ed. By, Cambridge University Press, Cambridge, UK. 1999

I have compiled my previous talks into a lecture series to give you flavor of my interests,

Lecture series on Epistemology, Philosophy, History and Chronology of the Ancient Indics and Their traditions .I plan to speak on Epistemic ruptures and Knowledge transmission in Hyderabad

[size="2"]L[color="#A0522D"]ECTURE I Introduction, Problem Of Indic Chronology, Why Study History (5)[/color]

[color="#006400"]Lecture II The Astronomical Heritage, Celestial Timekeepers (32)

Lecture III The Astronomical Heritage Part II (65)

Lecture IV Th astronomical Heritage Part III (79)[/color]

[color="#000080"]Lecture V ArchaeoAstronomy & Astrochronology (95)[/color]

[color="#8B0000"]Lecture VI Indic approach to creating knowledge (115)[/color]

[color="#FF8C00"]Lecture VII The Indic Savants (134)[/color]

[color="#800080"]Lecture VIII Sunya and Infinity (148)[/color]

[color="#FF00FF"]Lecture IX The Nature of the Mathematics (154)[/color]

[color="#00FF00"]Lecture X The transmission of Knowledge (172)[/size][/size][/size][/color]
Some 3 part Japanese documentary on Srinivasa Ramanujan:



Indian people had the great knowledge and learning power India's national language Hindi it also based on the mathematics that we divide and add the language and its is the world most developed language but it is not so popular...
Indian mathematician was the world best, they produced a lot of formulas and method we can see a glimpse of their effect on Indian language Hindi it is also based on the mathematics... so Indians got the great mathematicians till yet and they all made a great effect on science also...
nI am delighted to inform you that my book The Origins of astronomy will be available in various formats at LUlu and amazon. There is an author spotlight page where you can find the various formats

http://www.lulu.com/spotlight/kaushal44. I was truly a long drawn out ordeal to format the book.

There have been critiques of the book

1. That it is too big at 625 pages

2. Some [people sre frightened away by the Mathematics in the book (first year college level

3. That the book tries to cover too much

To quell the criticism arising from the first and second comment I will follow up with an Idiots guide to the Origins of astronomy. It is not a dumbed down version of the original. I will try to do an Issac Asimov type of book.Paradoxically the sentences that will replace the mathematical arguments will necessitate all my pedagogical skills that i possess, since mathematics is the best way to describe certain kinds of logic. I will have to more to say on this whole topic, so stay tuned.
i will be visiting,Delhi, Hyderabad and Visakha in December. This may prove to be the last time that i will travel extensively . I am planning to have a 1/2 day mini-conference in Delhi on Dec 10. I will announce the venue shortly. There will be half a dozen invitees who will lead of the discussion

If there are forum members in Delhi, Hyd or Vsakha I would be happy to to meet them especially if they hae an interest in the hIstory of the sciences anfd its impact on civlizations. if there are youngsters (in Delhi) interested in connecting the dots between seemingly unrelated events and topics , you might want to drop in or better yet volunteer to help in the conference . The topic for discussion will be

The Impact of a massive epistemic rupture and a distorted history on the future of the Indian civilization .

If you google my name you wil find ways of contacting me.

In my book, Ii have tried to answer the question whether there was a connection between the long period where EUROPE made little or no progress in the computational sciences between the time when of the Battle of Actium when Octavius Caesar defeated Mark Antony and Cleopatra defeating the last remnants of the EGYPTIAN Pharonic civilization and the fact that the Indian decimal place value made its way despite much opposition , thanks to the efforts of men like Simon Stevin. That the introduction of the decimal place value system let loose a veritable explosion in many subjects i have little doubt, but the question in my mind was WHY did it take EUROPE 1500 years to adopt such an obvious means of counting if it had been adopted so casually by the Indians as the west would have you believe. The Occident was emphatic in denying to india any kind of tradition that facilitated the adoption of such a system even in Vedic times and most Historian's in the occident have dismissed the invention of the DPV in ancient India, as an aberration(1) of which the Indians themselves did not have the slightest clue as to its real significance.

The answer was crystal clear. It had escaped the attention of the greatest scientists in the west , men like Archimedes , because it was a difficult concept to invent, especially for a people who were not used to numbers and who had to use multiplication tables to do even the simplest arithmetical operations.

Obversely (and this is where the lightbulb exploded ) it was not easy for the Indians either and they must have worked at it systematically over thousands of years beginning in the Vedic era to get to where they were during the time of Aryabhata where he makes use of these and other representations in such a facile manner . So comfortable was he with the entire edifice of what we call Number theory today that he developed the concept of a recursive algorithm while analytically developing the first sine difference table in the history of the world . Note that the use of recursion in an algorithm was considered a novelty and the sign of sophistication as late as the second half of the twentieth century. The most well known examples of Recursion are the Fibonacci sequence, the Kalman filter and the use of recursion in language.

But the use of recursion by Aryabhata was a significant indicator for me that the Indic tradition in the computational science was of high antiquity and very robust by the time of Aryabhata.

The name of the book is 'The origins of astronomy , the calendar and time ‘ The book is now available in many retail outlets including Amazon, . While it is not a demanding book , it requires an inquiring and restless mind that is not afraid of asking the hard questions. But for the person who reads the book in its entirety , I guarantee that it will change your view of the world. In my next post i will expain what i mean by an epistemic rupture

(1) quote from my book "The comment often made in the occident is that there is a general absence of proof in the ancient Indic texts. This reinforces the view that the Indic contributions were borrowed from elsewhere. Typical of such brashness was the remark of Morris Kline , May 1, 1908 – June 10, 1992 Professor Emeritus at Courant institute of Mathematical sciences‘ As our survey indicates the Hindus were interested in and contributed to the arithmetical and computational activities, rather than to the deductive pattern. Their name for mathematics is Gaṇita which means the Science of calculation. There is much good procedure and technical facility but no evidence that they considered Proof at all. They had rules but apparently no logical scruples. Moreover, no general methods or new viewpoints were arrived at in any area of mathematics.’ As if all this was not bad enough, he delivers the final coup de grace.

It is fairly certain that the Hindus did not appreciate the significance of their own contributions. The few good ideas that they had, such as separate symbols for the numbers from1 to 9 ’, the conversion from positional notation in base 60 to base 10, negative numbers and the recognition of 0 as a number , were introduced casually with no apparent realization that they were valuable innovations. They were not sensitive to mathematical values. The fine ideas that they themselves advanced, they commingled with the crudest ideas of the Egyptians and Babylonians."

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