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Vedic Mathematicians of Ancient India (part I)

 

Kosla Vepa Ph.D

 

 

 

This compendium of notes planned as a series of

 essays is dedicated to  the memory of Srinivasa

 Ramanujan (1887-1920), arguably the greatest

 number theorist in all of human history

Introductory Remarks

Uncovering the scope  of Ancient Indian Mathematics faces a twofold difficulty. To determine who discovered what we must have an accurate idea of the chronology of Ancient India.  This has been made doubly difficult by the faulty dating of  Indian Historical events by Sir William Jones, who practically invented the fields of linguistics and philology if for a moment we discount the contributions of Panini (Ashtadhyayi)and Yaska (Nirukta) a couple of millennia before him . Sir William, who was reputed to be an accomplished linguist, was nevertheless totally ignorant of Sanskrit  when he arrived in India and proceeded in short order to decipher the entire history of  India from his own meager understanding of the language, In the process he brushed aside the conventional history as known and memorized by Sanskrit pundits for hundreds  of years and as recorded in the Puranas and invented a brand new timeline for India which was not only egregiously wrong  but hopelessly scrambled up the sequence of events and personalities. See for instance my chronicle on the extent of the damage  caused by Sir William and his cohorts in my essay on the South Asia File  .

It is not clear whether this error was one caused by  inadequate knowledge of language or one due to deliberate falsification of records. It is horrific to think that a scholar of the stature of sir William would resort to skullduggery merely to satisfy his preconceived notions of the antiquity of Indic contributions to the sum of human knowledge. Hence we will assume Napoleon’s dictum was at play here and that we should attribute not to malice that which can be explained by sheer incompetence. This mistake has been compounded over the intervening decades by a succession of  British historians, who intent on reassuring themselves of their racial  superiority, refused to acknowledge the antiquity  of India, merely because ‘it could not possibly be’. When once they discovered the antiquity of Egypt, Mesopotamia and Babylon, every attempt was made not to disturb the notion that the Tigris Euphrates river valley was the cradle of civilization. When finally they stumbled upon increasing number of seals culminating in the discovery of Mohenjo Daro and Harappa by Rakhal Das Banerjee  and Daya Ram Sahni, they hit upon the ingenious idea that the Vedic civilization and the Indus Valley Civilization or the Saraswathi Sindhu Civilization, a more apt terminology since most of the archaeological sites lie along the banks of the dried up Saraswathi river, were entirely distinct and unrelated to each other. The consequences of such a postulate have been detailed in the  South Asia File.

The second difficulty was the Euro centricity(a euphemism  for a clearly racist attitude) of European mathematicians, who refused to appreciate the full scope of the Indic contributions and insisted on giving greater credit to Greece and later to Babylonian mathematics rather than recognize Indic and Vedic mathematics on its own merits. If this was indeed a surprise revelation, I fail to see the irony, when a similar Euro centricity was exhibited towards the antiquity of the Vedic people themselves.

The contributions of the ancient Indics are usually overlooked and rarely given sufficient credit in Western Texts (see for instance FAQ on Vedic Mathematics ).

The Wikipedia section on Indian Mathematics says the following;

Unfortunately, Indian contributions have not been given due acknowledgement in modern history, with many discoveries/inventions by Indian mathematicians now attributed to their western counterparts, due to Eurocentrism.

The historian Florian Cajori, one of the most celebrated historians of mathematics in the early 20th century, suggested that "Diophantus, the father of Greek algebra, got the first algebraic knowledge from India." This theory is supported by evidence of continuous contact between India and the Hellenistic world from the late 4th century BC, and earlier evidence that the eminent Greek mathematician Pythagoras visited India, which further 'throws open' the Eurocentric ideal.

More recently, evidence has been unearthed that reveals that the foundations of calculus were laid in India, at the Kerala School. Some allege that calculus and other mathematics of India were transmitted to Europe through the trade route from Kerala by traders and Jesuit missionaries. Kerala was in continuous contact with China, Arabia, and from around 1500, Europe as well, thus transmission would have

Furthermore, we cannot discuss Vedic mathematics without discussing Babylonian and Greek Mathematics to give it the scaffolding and context. We will devote some attention to these developments to put the Indic contribution in its proper context

However in recent years, there has been greater international recognition of the scope and breadth of the Ancient Indic contribution to the sum of human knowledge especially in some fields of science and technology such as Mathematics and Medicine. Typical of this new stance is the following excerpt by researchers at St. Andrews in Scotland.

An overview of Indian mathematics

It is without doubt that mathematics today owes a huge debt to the outstanding contributions made by Indian mathematicians over many hundreds of years. What is quite surprising is that there has been a reluctance to recognize this and one has to conclude that many famous historians of mathematics found what they expected to find, or perhaps even what they hoped to find, rather than to realize what was so clear in front of them.

We shall examine the contributions of Indian mathematics in this article, but before looking at this contribution in more detail we should say clearly that the "huge debt" is the beautiful number system invented by the Indians on which much of mathematical development has rested. Laplace put this with great clarity:-

The ingenious method of expressing every possible number using a set of ten symbols (each symbol having a place value and an absolute value) emerged in India. The idea seems so simple nowadays that its significance and profound importance is no longer appreciated. Its simplicity lies in the way it facilitated calculation and placed arithmetic foremost amongst useful inventions. The importance of this invention is more readily appreciated when one considers that it was beyond the two greatest men of Antiquity, Archimedes and Apollonius.

We shall look briefly at the Indian development of the place-value decimal system of numbers later in this article and in somewhat more detail in the separate article Indian numerals. First, however, we go back to the first evidence of mathematics developing in India.

Histories of Indian mathematics used to begin by describing the geometry contained in the Sulvasutras but research into the history of Indian mathematics has shown that the essentials of this geometry were older being contained in the altar constructions described in the Vedic mythology text the Shatapatha Brahmana and the Taittiriya Samhita. Also it has been shown that the study of mathematical astronomy in India goes back to at least the third millennium BC and mathematics and geometry must have existed to support this study in these ancient times.

Equally exhaustive in its treatment is the Wiki encyclopedia, where in general the dates are still suspect. See for instance the   Wikipedia on Indian Mathematics

 

Evidence From Europe That India Is The True  Birthplace Of  Our

 Numerals

 

The views of savants and learned scholars from a non-Indian tradition about Indian mathematics are presented here. Note that most of these are dated prior to the1800’s, when India was still untainted with the prefix of being a colonized country

Severus Sebokt of Syria in 662 CE: (the following statement must  be understood in the context of  the alleged Greek claim that all mathematical knowledge emanated from them

"I shall not speak here of the science of the Hindus, who are not even Syrians, and not of their subtle discoveries in astronomy that are more inventive than those of the Greeks and of the Babylonians; not of their eloquent ways of counting nor of their art of calculation, which cannot be described in words - I only want to mention those calculations that are done with nine numerals. If those who believe, because they speak Greek, that they have arrived at the limits of science, would read the Indian texts, they would be convinced, even if a little late in the day, that there are others who know something of value". (Nau, 1910)

Said al-Andalusi, probably the first historian of Science who in 1068 wrote Kitab Tabaqut al-Umam in Arabic  (Book of Categories of Nations) Translated into English by Alok Kumar in 1992

To their credit, the Indians have made great strides in the study of numbers (3) and of geometry. They have acquired immense information and reached the zenith in their knowledge of the movements of the stars (astronomy) and the secrets of the skies (astrology) as well as other mathematical studies. After all that, they have surpassed all the other peoples in their knowledge of medical science and the strengths of various drugs, the characteristics of compounds and the peculiarities of substances.

Albert Einstein in the 20th century also comments on the importance of Indian arithmetic: "We owe a lot to the Indians, who taught us how to count, without which no worthwhile scientific discovery could have been made."

Quotes from Liberabaci (Book of the Abacus) by Fibonacci (1170-1250): The nine Indian numerals are ...with these nine and with the sign 0 which in Arabic is sifr, any desired number can be written. (Fibonacci learnt about Indian numerals from his Arab teachers in North Africa) .Fibonacci introduced Indian numerals into Europe in 1202CE.

G Halstead

 
...The importance of the creation of the zero mark can never be exaggerated. This giving to airy nothing, not merely a local habituation and a name, a picture, a symbol but helpful power, is the characteristic of the Hindu race from whence it sprang. No single mathematical creation has been more potent for the general on go of intelligence and power. [CS, P 5]

The following quotes are from George Ifrah's book  Universal History

 of Numbers

    
The real inventors of this fundamental discovery, which is no less important than such feats as the mastery of fire, the development of agriculture, or the invention of the wheel, writing or the steam engine, were the mathematicians and astronomers of Indian civilisation: scholars who, unlike the Greeks, were concerned with practical applications and who were motivated by a kind of passion for both numbers and numerical calculations.

 
There is a great deal of evidence to support this fact, and even the Arabo-Muslim scholars themselves have often voiced their agreement


The following is a succession of historical accounts in favor of this theory, given in chronological order, beginning with the most recent

.
1. P. S. Laplace (1814): “The ingenious method of expressing every possible number using a set of ten symbols (each symbol having a place value and an absolute value) emerged in India. The idea seems so simple nowadays that its significance and profound importance is no longer appreciated. Its simplicity lies in the way it facilitated calculation and placed arithmetic foremost amongst useful inventions. The importance of this invention is more readily appreciated when one considers that it was beyond the two greatest men of Antiquity, Archimedes and Apollonius.” [Dantzig. p. 26]

 
2. J. F. Montucla (1798): “The ingenious number-system, which serves as the basis for modern arithmetic, was used by the Arabs long before it reached Europe. It would be a mistake, however, to believe that this invention is Arabic. There is a great deal of evidence, much of it provided by the Arabs themselves that this arithmetic originated in India.” [Montucla, I, p. 375J

 
3. John Walls (1616-1703) referred to the nine numerals as Indian figures [Wallis (1695), p. 10]


4. Cataneo (1546) le noue figure de gli Indi, “the nine figures from India”. [Smith and Karpinski (1911), p.3

 

 
5. Willichius (1540) talks of Zyphrae! Nice, “Indian figures”. [Smith and Karpinski (1911) p. 3]


6. The Crafte of Nombrynge (c. 1350), the oldest known English arithmetical tract: II fforthermore ye most vndirstonde that in this craft ben vsed teen figurys, as here bene writen for esampul 098 ^ 654321... in the quych we vse teen figwys of Inde. Questio II why Zen figurys of Inde? Soiucio. For as I have sayd afore thei werefondefrrst in Inde. [D. E. Smith (1909)

7. Petrus of Dada (1291) wrote a commentary on a work entitled Algorismus by Sacrobosco (John of Halifax, c. 1240), in which he says the following (which contains a mathematical error): Non enim omnis numerus per quascumquefiguras Indorum repraesentatur “Not every number can be represented in Indian figures”. [Curtze (1.897), p. 25

8.Around the year 1252, Byzantine monk Maximus Planudes (1260—1310) composed a work entitled Logistike Indike (“Indian Arithmetic”) in Greek, or even Psephophoria kata Indos (“The Indian way of counting”), where he explains the following: “There are only nine figures. These are:
123456789
[figures given in their Eastern Arabic form]

A sign known as tziphra can be added to these, which, according to the Indians, means ‘nothing’. The nine figures themselves are Indian, and tziphra is written thus: 0”. [B. N., Pans. Ancien Fonds grec, Ms 2428, f” 186 r”]

 
9. Around 1240, Alexandre de Ville-Dieu composed a manual in verse on written calculation (algorism). Its title was Carmen de Algorismo, and it began with the following two lines: Haec algorismus ars praesens dicitur, in qua Talibus Indorumfruimur bis quinquefiguris

 
Algorism is the art by which at present we use those Indian figures, which number two times five”. [Smith and Karpinski (1911), p. 11]


10. In 1202, Leonard of Pisa (known as Fibonacci), after voyages that took him to the Near East and Northern Africa, and in particular to Bejaia (now in Algeria), wrote a tract on arithmetic entitled Liber Abaci (“a tract about the abacus”), in which he explains the following:

 

 
Cum genitor meus a patria publicus scriba in duana bugee pro pisanis mercatoribus ad earn confluentibus preesset, me in pueritia mea ad se uenire faciens, inspecta utilitate el cornmoditate fiutura, ibi me studio abaci per aliquot dies stare uoluit et doceri. Vbi a mirabii magisterio in arte per nouem figuras Indorum introductus. . . Novem figurae Indorum hae sun!: cum his itaque novemfiguris. et turn hoc signo o. Quod arabice zephirum appellatur, scribitur qui libel numerus: “My father was a public scribe of Bejaia, where he worked for his country in Customs, defending the interests of Pisan merchants who made their fortune there. He made me learn how to use the abacus when I was still a child because he saw how I would benefit from this in later life. In this way I learned the art of counting using the nine Indian figures... The nine Indian figures are as follows:


987654321


[figures given in contemporary European cursive form].


“That is why, with these nine numerals, and with this sign 0, called zephirum in Arab, one writes all the numbers one wishes.”[Boncompagni (1857), vol.1]

11. C. U50, Rabbi Abraham Ben MeIr Ben Ezra (1092—1167), after a long voyage to the East and a period spent in Italy, wrote a work in Hebrew entitled: Sefer ha mispar (“Number Book”), where he explains the basic rules of written calculation.


He uses the first nine letters of the Hebrew alphabet to represent the nine units. He represents zero by a little circle and gives it the Hebrew name of galgal (“wheel”), or, more frequently, sfra (“void”) from the corresponding Arabic word.


However, all he did was adapt the Indian system to the first nine Hebrew letters (which he naturally had used since his childhood).

 
In the introduction, he provides some graphic variations of the figures, making it clear that they are of Indian origin, after having explained the place-value system: “That is how the learned men of India were able to represent any number using nine shapes which they fashioned themselves specifically to symbolize the nine units.” (Silberberg (1895), p.2: Smith and Ginsburg (1918): Steinschneider (1893)1


12. Around the same time, John of Seville began his Liberalgoarismi de practica arismetrice (“Book of Algoarismi on practical arithmetic”) with the following:


Numerus est unitatum cot/echo, quae qua in infinitum progredilur (multitudo enim crescit in infinitum), ideo a peritissimis Indis sub quibusdam regulis et certis lirnitibus infinita numerositas coarcatur, Ut de infinitis dfinita disciplina traderetur etfuga subtilium rerum sub alicuius artis certissima Jege ten eretur:

 “A number is a collection of units, and because the collection is infinite (for multiplication can continue indefinitely), the Indians ingeniously enclosed this infinite multiplicity within certain rules and limits so that infinity could be scientifically defined: these strict rules enabled them to pin down this subtle concept.
[B. N., Paris, Ms. lat. 16 202, p 51: Boncompagni (1857), vol. I, p. 261

13. C. 1143, Robert of Chester wrote a work entitled: Algoritmi de numero Indorum (“Algoritmi: Indian figures”), which is simply a translation of an Arabic work about Indian arithmetic. [Karpinski (1915); Wallis (1685). p. 121


14. C. 1140, Bishop Raymond of Toledo gave his patronage to a work written by the converted Jew Juan de Luna and archdeacon Domingo Gondisalvo: the Liber Algorismi de numero Indorum (“Book of Algorismi of Indian figures) which is simply a translation into a Spanish and Latin version of an Arabic tract on Indian arithmetic. [Boncompagni (1857), vol. 11


15. C. 1130, Adelard of Bath wrote a work entitled: Algoritmi de numero Indorum (“Algoritmi: of Indian figures”), which is simply a translation of an Arabic tract about Indian calculation. [Boncompagni (1857), vol. Ii


16. C. 1125, The Benedictine chronicler William of Malmesbury wrote De gestis regum Anglorum, in which he related that the Arabs adopted the Indian figures and transported them to the countries they conquered, particularly Spain. He goes on to explain that the monk Gerbert of Aurillac, who was to become Pope Sylvester II (who died in 1003) and who was immortalized for restoring sciences in Europe, studied in either Seville or Cordoba, where he learned about Indian figures and their uses and later contributed to their circulation in the Christian countries of the West. L Malmesbury (1596), f” 36 r’; Woepcke (1857), p. 35J


17. Written in 976 in the convent of Albelda (near the town of Logroño, in the north of Spain) by a monk named Vigila, the Coda Vigilanus contains the nine numerals in question, but not zero. The scribe clearly indicates in the text that the figures are of Indian origin:


Item de figuels aritmetice. Scire debemus Indos subtilissimum ingenium habere et ceteras gentes eis in arithmetica et geometrica et ceteris liberalibu.c disciplinis concedere. Et hoc manifèstum at in novem figuris, quibus quibus designant unum quenque gradum cuiu.slibetgradus. Quatrum hec sunt forma:


9 8 7 6 5 4 3 2 1.


“The same applies to arithmetical figures. It should be noted that the Indians have an extremely subtle intelligence, and when it comes to arithmetic, geometry and other such advanced disciplines, other ideas must make way for theirs. The best proof of this is the nine figures with which they represent each number no matter how high. This is how the figures look:

 
9 8 7 6 5 4 3 2 1

 

Al-Khwarismi (783-850 CE) Popularized Indian numerals, mathematics  including Algebra in the Islamic world  and the Christian West .Algebra was named after his treatise 'Al jabr wa'l Muqabalah'' which when translated from Arabic means 'Transposition and Reduction'. Little  is known about his life except that he lived at the court of the Abbasid Caliph al Ma'amun , in Baghdad shortly after  Charlemagne was made emperor of the west. and that he was one of the most important mathematicians and astronomers who worked at the house of Wisdom (Bayt al Hikma)

'

Muhammad Ben Musa aI-Khuwarizmi (circa 783—850.).

 Portrait on wood made in 1983 from a Persian illuminated

 manuscript for the l200th anniversary of his birth. Museum

of the Ulugh Begh Observatory. Urgentsch (Kharezm).

 Uzbekistan (ex USSR). By calling one of its fundamental

 practices and theoretical activities the algorithm computer

 science commemorates this great Muslim scholar.  

 

 

 

Links

FAQ On the Mathematics of the Vedics

Does no one remember the Hindu contribution to Mathematics?

Mathematics in ancient India

A sample of vedic mathematics

Ancient Indian Mathematics

 

Indian Mathematics

"The first mathematics which we shall describe in this article developed in the Indus valley. The earliest known urban Indian culture was first identified in 1921 at Harappa in the Punjab and then, one year later, at Mohenjo-Daro, near the Indus River in the Sindh. Both these sites are now in Pakistan but this is still covered by our term "Indian mathematics" which, in this article, refers to mathematics developed in the Indian subcontinent. The Indus civilisation (or Harappan civilisation as it is sometimes known) was based in these two cities and also in over a hundred small towns and villages. It was a civilisation which began around 2500 BC and survived until 1700 BC or later. The people were literate and used a written script containing around 500 characters which some have claimed to have deciphered but, being far from clear that this is the case, much research remains to be done before a full appreciation of the mathematical achievements of this ancient civilisation can be fully assessed. "

The above statement must be revised based on new archaeological discoveries. More than 400 sites have been found along the banks of the dried up river bed of the ancient river Saraswathi. These sites include the submerged city of Bet Dwaraka, the city ruled by Sri Krishna during the episodes of the Mahabharata and the great Bharata war that is described in detail in that epic. The important point to note is that a prerequisite to do numerical work is a script. So, there must have been a script by the time the Saraswathi Sindhu civilization was flourishing not just centered in the two cities of Mohenjo Daro and Harappa but along dozens of urban towns and cities like Dholavira, Lothal, Dwaraka and others. European historians often wonder what happened to the  denizens of the Indus Valley civilization. Ockham’s razor suggests the right answer . Nothing catastrophic happened to these people and we the modern Indics are the descendants of this civilization which was spread over a huge area stretching from Haryana in the north to the present day province of Maharashtra to places like Prathishtan (later Pathan) which eventually became the capital of   the Satavahana Kingdoms are in fact a successor to the Urban civilizations that existed prior to them. This makes eminent sense because the word Brahmi signifies the goddess Saraswathi (consort of Brahma) and is therefore also considered to be the Guardian deity of Knowledge and the one who is credited with blessing us with the gift of a script. There are a group of  Brahmanas in the Konkan area of present day state of Karnataka  who call themselves Saraswath Brahmanas and legend has it that they migrated from the banks of the Saraswath river when it eventually dried out. In fact  the Gowda Saraswath Brahmanas   have done extremely well over the succeeding centuries and have prospered far in excess of their proportion in the population. In fact our family records show that about 15 generations ago my ancestor by  the name of Hanuman Bhat migrated to the Andhra country , to escape the  turmoil caused by the interminable wars and the tyranny of Aurangazeb , from the area which is considered present day Konkan

We often think of Egyptians and Babylonians as being the height of civilisation and of mathematical skills around the period of the Indus civilisation, yet V G Childe in New Light on the Most Ancient East (1952) wrote:-

India confronts Egypt and Babylonia by the 3rd millennium with a thoroughly individual and independent civilisation of her own, technically the peer of the rest. And plainly it is deeply rooted in Indian soil. The Indus civilisation represents a very perfect adjustment of human life to a specific environment. And it has endured; it is already specifically Indian and forms the basis of modern Indian culture.

The Sutra Era of Vedic Mathematics

 Notes

That the contributions of the Indics were considerable was therefore in little doubt among the Europeans in the middle ages . It is only when we  come to the colonial era that the British had a reluctance to admit this glaring fact. In the sequel to this essay we will lay out  the chronology of these discoveries in the mists of a bygone era.

next

Vedic mathematics in Ancient India Part II

 

 

 

 

 

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