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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.
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