Vedic Mathematicians in Ancient India (PartI)

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Arial;">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.

font-family: Arial;">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

0);">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

-moz-initial; -moz-background-inline-policy: -moz-initial;">. The historian Florian Cajori, one

of the most celebrated historians of mathematics in the early 20th century,

suggested that "Diophantus

-moz-initial; -moz-background-origin: -moz-initial;

-moz-background-inline-policy: -moz-initial;">, the father of Greek algebra,

got the first algebraic knowledge from India." This theory is supported by

evidence of continuous contact between

color: rgb(153, 51, 0);">India and the Hellenistic world from the late

-moz-initial; -moz-background-inline-policy: -moz-initial; color: rgb(153, 51,

0);">4th century BC, and earlier evidence that the eminent Greek mathematician

scroll 0% 50%; -moz-background-clip: -moz-initial; -moz-background-origin:

-moz-initial; -moz-background-inline-policy: -moz-initial; color: rgb(153, 51,

0);">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

-moz-background-inline-policy: -moz-initial;">. Some allege that calculus and

other mathematics of India were transmitted to

-moz-initial; color: rgb(153, 51, 0);">Europe through the trade route from Kerala by traders and

-moz-background-inline-policy: -moz-initial; color: rgb(153, 51, 0);">Jesuit

missionaries. Kerala was in continuous contact with China,

-moz-background-origin: -moz-initial; -moz-background-inline-policy:

-moz-initial; color: rgb(153, 51, 0);">Arabia, and from around 1500, Europe as

well, thus transmission would have

font-family: Arial;">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:-

-moz-initial; color: maroon; font-family: Arial;">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

maroon;">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)

text-align: justify;">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

(1616-4703)

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]

15.75pt;

text-align: justify;">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

15.75pt; text-align: justify;">[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).

Arial;">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:

justify;">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 Logro�o, 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 manif�stum at in novem figuris, quibus quibus

designant unum quenque gradum cuiu.slibetgradus. Quatrum hec sunt forma:

font-family: Arial;">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

text-align: justify;"> 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.

15.75pt; text-align: justify;"> 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.

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