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ॐ पूर्णमदः पूर्णमिदम् पूर्णात्
पूर्णमुदच्यते |
पूर्णस्य पूर्णमादाय
पूर्णमेवावशिष्यते ||
ॐ शान्तिः, शान्तिः, शान्तिः ||
Oum poorNamadaH poorNamidam poorNaat poorNamudachyate,
poorNasya poorNamaadaaya
poorNamevaavaSiShyate,
Oum SaantiH, SaantiH, SaantiH.
That is Absolute, This is Absolute, Absolute arises out of
Absolute, If Absolute is taken away from Absolute, Absolute
remains
OM Peace, Peace, Peace.
This section is dedicated to the
memory of
Srinivasa Ramanujan
(1887-1920), the man who knew infinity, arguably the greatest number theorist in all of
human history
Introductory
Remarks
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 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
Vedic
number theory
Mathematicians
in
India
were interested in finding integral
solutions of
Diophantine
equations
since the
Vedic
era. The earliest geometric use of
Diophantine equations can be traced
back to the
Sulba Sutras,
which were written between the 8th
and 6th centuries BC.
Baudhayana
(c. 800 BC) found two sets of
positive integral solutions to a set
of simultaneous Diophantine
equations, and also used
simultaneous Diophantine equations
with up to four unknowns.
Apastamba
(c. 600 BC) used simultaneous
Diophantine equations with up to
five unknowns.
Jaina
number theory
In India,
Jaina
mathematicians developed the
earliest systematic theory of
numbers from the 4th century BC to
the 2nd century CE. The Jaina text
Surya Prajinapti (c. 400 BC)
classifies all numbers into three
sets: enumerable, innumerable and
infinite. Each of these was further
subdivided into three orders:
-
Enumerable: lowest,
intermediate and highest.
-
Innumerable: nearly
innumerable, truly innumerable
and innumerably innumerable.
-
Infinite: nearly infinite,
truly infinite, infinitely
infinite.
The Jains were
the first to discard the idea that
all infinites were the same or
equal. They recognized five
different types of infinity:
infinite in one and two directions
(one
dimension),
infinite in area (two dimensions),
infinite everywhere (three
dimensions), and infinite
perpetually (infinite number of
dimensions).
The highest
enumerable number N of the
Jains corresponds to the modern
concept of
aleph-null
(the
cardinal
number
of the infinite set of integers 1,
2, ...), the smallest cardinal
transfinite
number.
The Jains also defined a whole
system of transfinite cardinal
numbers, of which
is the smallest.
In the Jaina
work on the
theory of sets,
two basic types of transfinite
numbers are distinguished. On both
physical and
ontological
grounds, a distinction was made
between asmkhyata and
ananata, between rigidly bounded
and loosely bounded infinities.
Hellenistic number theory
Number theory
was a favorite study among the
Hellenistic
mathematicians
of
Alexandria,
Egypt
from the 3rd century CE, who were
aware of the
Diophantine
equation
concept in numerous special cases.
The first Hellenistic mathematician
to study these equations was
Diophantus.
Diophantus
also looked for a method of finding
integer solutions to
linear
indeterminate
equations,
equations that lack sufficient
information to produce a single
discrete set of answers. The
equation x
+ y = 5 is such an
equation. Diophantus discovered that
many indeterminate equations can be
reduced to a form where a certain
category of answers is known even
though a specific answer is not.
Classical
Indian number theory
Diophantine
equations were extensively studied
by mathematicians in medieval India,
who were the first to systematically
investigate methods for the
determination of integral solutions
of Diophantine equations.
Aryabhata
(499) gave the first explicit
description of the general integral
solution of the linear Diophantine
equation ay
+ bx
= c, which occurs in his text
Aryabhatiya. This kuttaka
algorithm is considered to be one of
the most significant contributions
of Aryabhata in pure mathematics,
which found solutions to Diophantine
equations by means of
continued
fractions.
The technique was applied by
Aryabhata to give integral solutions
of simulataneous linear Diophantine
equations, a problem with important
applications in astronomy. He also
found the general solution to the
indeterminate
linear
equation
using this method.
Brahmagupta
in 628 handled more difficult
Diophantine equations. He used the
chakravala
method
to solve
quadratic
Diophantine equations, including
forms of
Pell's
equation,
such as 61x2
+ 1 = y2.
His
Brahma
Sphuta Siddhanta
was translated into
Arabic
in 773 and was subsequently
translated into
Latin
in 1126. The equation
61x2
+ 1 = y2
was later posed as a problem in 1657
by the
French
mathematician
Pierre de
Fermat.
The general solution to this
particular form of Pell's equation
was found over 70 years later by
Leonhard Euler,
while the general solution to Pell's
equation was found over 100 years
later by
Joseph Louis
Lagrange
in 1767. Meanwhile, many centuries
ago, the general solution to Pell's
equation was recorded by
Bhaskara II
in 1150, using a modified version of
Brahmagupta's chakravala
method, which he also used to find
the general solution to other
indeterminate quadratic equations
and quadratic Diophantine equations.
Bhaskara's chakravala method
for finding the general solution to
Pell's equation was much simpler
than the method used by Lagrange
over 600 years later. Bhaskara also
found solutions to other
indeterminate quadratic,
cubic,
quartic
and higher-order
polynomial
equations.
Narayana
Pandit
further improved on the
chakravala method and found more
general solutions to other
indeterminate quadratic and
higher-order polynomial equations.
EVIDENCE PROM EUROPE
INDIA: THE TRUE BIRTHPLACE OF
OUR NUMERALS
The views of savants
and learned scholars from a non-Indian tradition about Indian
mathematics
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 (179$): 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.31
5. Willichius (1540) talks of Zyphrae
!Indicate,
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)1
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. 251
8.Around the year 1252, Byzantine monk Maximus
Planudes (12601310) 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 zephi rum in Arab,
one writes all the numbers one
wishes.[Boncompagni (1857), vol.1]
11. C. U50, Rabbi Abraham Ben MeIr Ben Ezra
(10921167), 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 symbolise
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, r 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
trans lation of an Arabic work about Indian
arithmetic. [Karpinski (1915); Wallis (1685). p.
121
14. C. 1140, Bishop Raimundo of Toledo gave his
patronage to a work written by the converted jew
Juan de 1.una 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; Kopeck (1857), p. 35J
17. Written in 976 in the convent of Albelda
(near the town of Logrońo, 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 manifčstum 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.
'
Book Reviews
The Origins of Mathematics
What India should Know
What
India Should Know - A book
review
The distortion of
Indias past by western
historians
V.
Lakshmikantham & J.
Vasundhara Devi;
What India Should Know,
Bharatiya Vidya Bhavan,
pp 308, Rs 250.00
By Manju Gupta
The deep-rooted
prejudices about the
qualities, traditions
and religions of the
East have been a
pervasive and marked
characteristic of
Western thought of
centuries. It was a
thought reinforced in
the 19th century by
industrialisation and
imperialism, and which
resulted in
identification of the
East with backwardness
and ungovernability.
We also agree that today
scholarship means being
at home with what is
written by Western
scholars, who have more
than often discredited
the ancient past of
Indian culture and
distorted the history
and chronology of
events.
The book under review,
written by
mathematicians Dr V.
Lakshmikantham and Dr J.
Vasundhara Devi, begins
by throwing light on the
confusion till today
between Gupta
Chandragupta and Maurya
Chandragupta. They point
out that actually Gupta
Chandragupta flourished
in 327 BC and was the
contemporary of
Alexander, while Maurya
Chandragupta lived in
1534 BC. But the
Western historians
wrongly identified
Alexanders contemporary
with Maurya
Chandragupta, thus
affecting more than
1,200 years in the
history of ancient
India. This colossal
blunder upset the whole
scheme and brought
terrible chaos into the
Puranic dates of India.
They point out that it
was Sir William Jones,
the first historian of
India, who changed this
date to effect a sort of
similitude between the
Biblical and Indian
conceptions of time and
they add, twelve
centuries of time after
the Mahabharata war
(3138 BC) and 10
centuries before that
are struck off like this
and the history the
Indians got to know is
put upon this wrong
base. The Western
scholars have not only
bungled facts and
tampered with texts, but
even gone to the extent
to hurling abuse at
ancient Indian
historians and sages.
The authors feel that
colonisation had
affected the Indian mind
in certain aspects.
Through Macaulays
education policies, the
British ensured that
they left behind an
inferiority complex
among the Indians by
constantly denigrating
Indian culture. This is
why the intellectuals of
India today repeat what
their masters said
before and ape them
after having hated
them, say the authors.
They add that another
masterstroke of the
British was the
propagation of the
absurd theory of Aryan
invasion according to
which India was invaded
by a tribe called Aryans
who originated in
western Russia and
imposed upon the
Dravidians of India, the
hateful caste system.
They continue, To the
Aryans are attributed
Sanskrit, the Vedic
religion, as well as
Indias greatest
spiritual texts, the
Vedas and a host of
writings like the
Upanishads. The Aryan
invasion myth has shown
that the Indian
civilisation was not
that ancient and that it
was secondary to the
cultures that influenced
the Western world. Also,
whatever good thing
India had developed has
been a consequence of
the influence of the
West.
The book deals with the
general prejudice about
the East, the distortion
of Indian history and
the superficial
translation of the Vedas
by Western scholars. The
authors comment
ironically that the
supposedly enlightened
writers such as Edward
Gibbon who never set
foot east of
Switzerland, in his
History of the Roman
Empire, loved to make
play of the despicable
people of the East, and
Voltaire, who never
travelled beyond Berlin,
fantasised about the
misery and bigotry of
the Eastern nation.
They add, The most
conspicuous example was
Lord Macaulay, who
carried his
all-consuming racist
hatred of the East to
ridiculous depths by
asserting that the
entire corpus of
knowledge that the
Orient possessed could
be contained in half a
thimble. They add that
the world is but one and
the East and West
bifurcation is a
mythical boundary.
The catastrophic event
of the formation of a
Mediterranean Sea
resulted in the loss of
culture and civilisation
existing in Europe. The
history of the Greeks,
Roman and the British
are traced briefly and
so is the awakening of
Europe from the dark
ages.
The book ridicules the
theory of Aryan invasion
and gives in points the
reasons for its
dismissal. It says that
the Aryans spread from
the Bharatavarsha in
different directions to
spread the Aryan
culture. There was
never any Aryan invasion
of India or any
Aryan-Dravidian war. The
cradle of civilisation
was not Sumeria in
Mesopotamia, but the
Sapta Sindhu, the land
of seven rivers in
north-west India.
Then it expounds on the
misrepresentation of the
two Chandraguptas and
tries to set right the
chronology of events in
India.
It points out that the
Aryan invasion theory
was aimed at dividing
India into factions. It
explains that the Aryans
were extremely sensitive
to the high walks of
life, righteousness and
nobility, both in
thought and action. That
is, the Aryans followed
the Vedic Dharma, also
called the Sanatana
Dharma. Dharma is that
nature which makes a
thing what it is. Thus
Manava Dharma implies
that human beings
should be true to their
own essential nature,
which is divine;
therefore, all efforts
in life should be
directed towards
maintaining the dignity
of the atma (the self)
and not plodding through
life like helpless
animals. Thus Dharma is
the law of being.
The book exposes the
deliberate distortions
wrought by Orientalists
in their efforts to
write the history of
India.
The book traces the
great traditions laid
down by Sanatana Dharma
throughout the world
that endured in
Bharatakhand in the 12th
century.
And the authors try to
synthesise India with
its glorious heritage
and the present
technological advances
ready for taking India
into the twenty-first
century. The chapter
ends on a positive note
that this entry will
have a new awakening and
the humanity will be
much more spiritual than
it has been.
The book concludes by
saying that the Sanatana
Dharma is much more
open than any other
religion to new ideas,
scientific thought and
social experimentation.
Many principles basic to
Sanatana Dharma
initially appeared
strange to the West,
such as yoga,
meditation,
reincarnation and
methods of
interiorisation, but
these principles have
now found worldwide
acceptance. Sanatana
Dharma is, of course, a
world religion
(Bharatiya Vidya Bhavan,
Kulpati Munshi Marg,
Mumbai - 400 007.)
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http://www.archaeologyonline.net/artifacts/mankind-invention.html
MANKINDS GREATEST INVENTION
The story of the search for
the perfect number system
By N.S. Rajaram
The Universal History of
Numbers, 3 volumes by
Georges Ifrah (2005 PB).
Penguin, India.
Finally it all came to pass
as though across the ages
and the civilizations, the
human mind had tried all the
possible solutions to the
problem of writing numbers,
before universally adopting
the one which seemed the
most abstract, the most
perfected and the most
effective of all.
In these memorable words,
the French-Moroccan scholar
Georges Ifrah, the author of
the monumental but somewhat
flawed The Universal History
of Numbers, sums up the many
false starts by many
civilizations until the
Indians hit upon a method of
doing arithmetic which
surpassed and supplanted all
others one without which
science, technology and
everything else that we take
for granted would be
impossible. This was the
positional or the place
value number system. It is
without a doubt the greatest
mathematical discovery ever
made, and arguably Indias
greatest contribution to
civilization.
The three-volume Indian
edition is the English
version of the 1994 French
edition. It tells the story
of humanitys 3000-year
struggle to solve the most
basic and yet the most
important mathematical
problem of all counting.
The first two volumes
recount the tortuous history
of the long search that
culminated in the discovery
in India of the modern
system and its westward
diffusion through the Arabs.
The third volume, on the
evolution of modern
computers, is not on the
same level as the first two.
Better accounts exist.
The term Arabic numerals
is a misnomer; the Arabs
always called them Hindi
numerals. What is remarkable
is the relatively
unimportant role played by
the Greeks. They were poor
at arithmetic and came
nowhere near matching the
Indians. Babylonians a
thousand years before them
were more creative, and the
Maya of pre-Colombian
America far surpassed them
in both computation and
astronomy. The Greek Miracle
is a modern European
fantasy.
The discovery of the
positional number system is
a defining event in history,
like mans discovery of
fire. It changed the terms
of human existence. While
the invention of writing by
several civilizations was
also of momentous
consequence, no writing
system ever attained the
universality and the
perfection of the positional
number system. Today, in the
age of computers and the
information revolution,
computer code has all but
replaced writing and even
pictures. This would be
impossible without the
Indian number system, which
is now virtually the
universal alphabet as well.
What makes the positional
system perfect is the
synthesis of three simple
yet profound ideas: zero as
a numerical symbol; zero
having nothing as its
value; and the zero as a
position in a number string.
Other civilizations,
including the Babylonian and
the Maya, discovered one or
other feature but failed to
achieve the grand synthesis
that gave us the modern
system. Of the worlds
civilizations, the Mayas
came closest. They, like the
Babylonians, had an idea of
the zero, but never learnt
how to operate with it.
In Ifrahs words: The
measure of genius of the
Indian civilization, to
which we owe our modern,
system, is all the greater
in that it was the only one
in all history to have
achieved this triumph.
Modern civilization rests on
the modern number system.
The decimal system is just a
special case of it.
The synthesis was possible
due to the Indians capacity
for abstract thought: they
saw numbers not as visual
aids to counting, but as
abstract symbols. While
other number systems, like
the Roman numerals for
example, expressed numbers
visually, Indians early
broke free of this shackle
and saw numbers as pure
symbols with values. We see
it in other fields also. The
great grammarian Panini
describes the Indian
alphabet in purely phonetic
terms, without reference to
symbols. It is the same in
music. While the Western
notation depends on both the
form and the location of
notes written across staves,
the Indian notation can use
any seven symbols.
The economy and precision of
the positional system has
made all others obsolete.
Some systems could be
marvels of ingenuity, but
led to incredible
complexities. The Egyptian
hieroglyphic system needed
27 symbols to write a number
like 7659. Another
indispensable feature of the
Indian system is its
uniqueness. Once written, it
has a single value no matter
who reads it. This was not
always the case with other
systems. In one Maya
example, the same signs can
be read as either 4399 or
4879. It was even worse in
the Babylonian system, where
a particular number string
can have a value ranging
from 1538 to a fraction less
than one! So a team of
scribes had be on hand to
cross check numbers for
accuracy as well as
interpretation.
The Universal History of
Numbers is an impressive
achievement but not a
definitive work. It has
several drawbacks errors of
omission and commission that
are perhaps unavoidable when
one tries to cover a vast
area spanning space, time
and civilizations. The
authors discussion of
palaeography sometimes goes
awry due to his reliance on
secondary sources, some of
which go back to the
nineteenth century. He
accepts as proven
conclusions that are
contentious and even
demonstrably false. (Like
his acceptance of the
non-existent Aramaeo-Brahmi
as the source of the Brahmi
alphabet.) These, however,
do not seriously detract
from a marvelous work. The
books may be read by anyone
with an interest in
mathematics.
In summary, Georges Ifrah
has opened the gates for
what promises to be a major
new pathway for research. It
is now for others to rise to
the challenge.
_________
N.S. Rajaram is a
mathematician who has
written on ancient history.
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Indias
scientific
tradition
By Amba
Charan
Vashishth
Suresh Soni:
Indias
Glorious
Scientific
Tradition:
Ocean Books
Pvt. Ltd.;
pp. 248; Rs.
300.00
The Muslim
and the
British
invaders
hammered
home the
message in
the Indian
mind that
their
religion,
language,
literature,
culture and
philosophy
were
superior to
that of
India and
everything
Indian was
inferior and
useless. To
a great
extent they
succeeded
too. It
instilled in
us an
inferiority
complex in
whatever was
Indian. Our
winning
freedom was
just a moral
victory over
the English
but it
failed to be
a real
victory over
our slavish
sense of
inferiority
in
ourselves.
Yet,
regrettably,
it failed to
instil in
our minds a
sense of
self-pride.
The worst
was the
field of
science. It
was
repeatedly
said that
the sun of
scientific
knowledge
rose in the
West from
where we got
the first
rays through
a
reflection.
But with his
scholarly
exposition,
in his
latest book,
Indias
Glorious
Scientific
Tradition,
the Sah
Sarkaryavah
of RSS Shri
Suresh Soni
has smashed
the myth
that the
first rays
of science
broke out in
the West and
thus started
the wheel of
development
throughout
the world.
He regrets
that this
unawareness
of the fact
that we had
a scientific
tradition
and a
scientific
point of
view,
resulted in
the lack of
faith that
we could
have a role
in todays
world.
The main
reason seems
to be that
the treasure
of knowledge
(included
science) of
India lay in
our Vedas,
Upnishads,
Smritis,
shrutis,
shlokas,
classics
Ramayana,
Mahabharata
and other
literature.
These were
made to
appear as if
these
contained
only Hindu
philosophy
and
ideology.
These
classics had
no tag of
science. In
fact,
scientific
knowledge
formed the
very part
and parcel
of our
philosophy
and dharma
trove. All
this
knowledge,
for our
invader-rulers,
was only
religious
and,
therefore,
for them
detestable.
They never
tried to
study. They
never
believed it.
For them it
was only
heresy,
imaginary
and
unbelievable.
They only
tried to
impose their
piece of
knowledge,
their
literature
and that
philosophy.
Those who
got
education
under the
system of
education
introduced
by aliens in
non-Hindu
stream of
thought got
swayed by
the alien
thought and
became their
vocal
champions.
Shri Soni
has
highlighted
the fact
that in the
beginning of
20th
century,
many
scholars,
like Acharya
Prafulla
Chandra Rai,
Brajendra
Nath Seel,
Jagdish
Chandra Basu,
Rao Saheb
Vaze, to
name only a
few, had
proved that
from the
earliest
times of
civilisation
not only in
religious
philosophies,
even in
science and
technology
India was
second to
none.
In ancient
India the
quest for
knowledge
was
unending,
unfettered.
In India no
Bruno had to
be burnt
alive and no
John Kepler
had to die
for not
apologising
to Pope for
his
scientific
discovery
which the
latter did
not approve.
Here in
India
Charvak said
those who
created the
Vedas were
hypocrites,
cunning and
evil was
never
maltreated,
yet he was
not
persecuted
but was
accorded the
status of a
philosopher.
The writer
has detailed
the
scientific
knowledge
and acumen
through
presentation
of
information
in different
chapters of
the book
devoted to
Indian
Concept of
Science,
Electrical
and
Mechanical
Science,
Metallurgy,
Aeronautics,
Measurement
of Time,
Astronomy,
Architecture,
Chemistry,
Botany,
Zoology and
others.
Scientific
tradition,
according to
the writer,
is not
something
new; it is
as old as
the Indian
civilisation
itself. Sage
Bhrigu
mentions ten
shastrasagriculture,
hydrology,
mining,
shipping,
charioteering,
rocket
science,
Weshm
Shastra
(house,
buildings
etc.),
Praakar
Shaastra,
town
planning and
mechanical
engineering,
etc.
Metallurgy
formed part
of Ayurveda
and ancient
Indian
physicians
like Charak,
Sushruta and
Nagarjuna
have
described in
detail how
to prepare
medicines
from gold,
silver,
copper,
iron, mica,
mercy, etc.
Qutab Minar
is another
example of
Indias
excellence
in
metallurgy.
Al Baruni
makes a
mention of
it in 11th
century.
Qutab Minar
was made in
the 4th
century
it
is also
called the
Garuda
pillar. It
was brought
to Delhi in
1050 by
Anang Pal,
the founder
of Delhi.
Sage
Bharadwaj
had authored
Yantra
Sarvaswa one
part of
which deals
with
aeronautical
science (Vaimanik
Shastra). He
mentions six
aeronautics
scholars
before him,
defines a
plane and
speaks of 32
secrets to
fly a plane
and mentions
four kinds
of energy
sources
needed to
fly an
aeroplane.
Sage
Shaunaka
divides the
airways in
five
different
ways while
Dhundinath
speaks of
the
different
whirlpools
on the
heights of
various
paths and
then gives
an
indication
of hundreds
of paths on
each of
those
heights.
Viman
Shastra
speaks of 31
kinds of
machines and
their
specific
places in
the plane.
Referring to
the strength
of Indias
flourishing
economy,
Shri Soni
quotes
Samuel
Huntingtons
book, The
Clash of
Civilisations,
that in 1750
Indias
production
was 24.5 per
cent, while
Europes was
18.2 per
cent and
that of the
Soviet Union
was 5 per
cent in the
world. He
quotes the
President Dr
APJ Abdul
Kalam that
the British
admitted
that rockets
used by Tipu
Sultan were
the first
ones to be
used in war
in the world
against the
British
army.
Cotton
plant,
cotton
fibres,
thread made
of it and
the cloth
were all new
to the
British.
Hence, they
used to say
that Indians
were very
clever. They
grow the
wool on
plants that
ought to be
on the sheep
and weave
cloth with
it. Traders
from abroad
started
ordering
cotton cloth
from India,
especially
the mulmul
from Bengal.
The British
destroyed
this cottage
industry and
cut off the
thumbs that
made such
fine cloth.
He says
electrical
science is
very ancient
in India and
quotes
Agastya
Samhita
which
states: Take
an earthen
pot, place
of copper
sheet, and
put the
shikhigreeva
in it. Then,
smear it
with west
sawdust,
mercury and
zinc. Then,
if you join
the wires,
it will give
rise to
Mitravarunashakti.
In
Samarangan
Sutradharit
is stated:
The mass
and momentum
of flowing
water
current is
used in
hydraulic
machines or
turbines for
the
generation
of power.
The writer
quotes
Agastya
Samhita to
highlight
Indias
advancement
in
electricity
generation
and
electroplating.
Sage
Kanaads
Vaisheshik
Darshan
holds mirror
to Indias
ancient
mechanical
science.
During
Chalukyas
reign India
had a
self-operational
system of
draining
water from a
tank.
Navigation
Indian
Navys motto
reads: Sham
No Varunah,
meaning
Lord of the
water be
compassionate
or kind to
us.
Information
on
manufacturing
of ships can
be found in
Brihat
Samhita,
written by
Varahmihir
in 5th
century and
Yukti
Kalpataru by
Raja Bhoj in
11th
century. A
foreign
traveler
named Nicolo
Conti who
visited
India in
15th century
writes, The
Indian ships
are much
bigger than
our ships.
In the early
19th century
Dr. Taylor
writes,
When the
Indian ships
laden with
Indian goods
reached the
port of
London, it
created such
a panic
amongst the
British
traders as
would not
have been
created, had
they seen
the enemy
fleet of
ships on the
River
Thames,
ready for
attack.
According to
famous
archeologist
Padmashri Dr
Vishnu
Shridhar
Wakankar,
Vasco da
Gama had not
discovered
India but he
came
following a
Gujarati
navigator
Chandan from
near
Zanzibar in
Africa.
In India,
Mathematics
is as old as
the Vedas. A
Yajush
jyotisham
shloka says
just as the
crest of the
peacock and
the gem of
the cobra
stays right
at the top,
similarly
mathematics
is
established
right at the
topmost
position in
the Vedas
and the
shastras.
We have two
kinds of
thinkersiti,
the one who
said
something
about
completeness
and neti who
said
something
about the
zero. In his
book The
Foundation
and Process
of
Mathematics,
Prof. G. P.
Halstand
says, The
significance
or
importance
of the
discovery of
zero can
never be
explained.
Giving not
just a name
but
authority,
in fact,
power to
nothing,
is the
characteristic
of the Hindu
community,
whose
invention,
it is.
Decimal
system too
is Indias
contribution.
The oldest
European
book on
mathematics
Coda
Vigilanus
kept in
Museum at
Madrid
(Spain)
says: From
the signs of
counting
(numerals),
we
experience
that the
ancient
Hindus had
very sharp
brains and
that the
other
countries
were much
behind them
in counting
and in
geometry and
others
sciences.
Algebra and
Geometry had
their origin
in India.
Bodhayan had
described
1000 years
earlier what
Pythagoras
stated later
and came to
be known as
Pythagoras
Theorem.
Based on
Bodhayan
Theorem the
principles
of
trigonometry
have
naturally
been given
in the
Shulbhasootra
Vedic
Mathematics
Shankaracharya
Bharati
Krishna
Teerth of
Puri
discovered a
new
mathematical
process,
called
Mathematics
Without
Tears. He
also called
it Vedic
Mathematics.
This was
unbelievable
for European
mathematicians.
When the
Shankaracharya
gave a live
demonstration,
Prof.
Nicholas of
England
called it
magic and
not mere
mathematics.
When asked
to explain,
Teerathji
said, until
you do not
know it is
magic; when
you get to
know it, it
is
mathematics.
Vedic
Mathematics
has now come
to be taught
for
competitive
examinations.
In the
science of
measurement
of time too,
India was
much far
ahead of all
others. The
famous
astronomer
Bally of
Europe says,
According
to the
planetary
calculations
of the
Hindus, the
present age,
that is
Kaliyug,
started
3,102 years
ago on the
20th of
February at
2 hours 27
minutes and
30 seconds.
As such,
these
calculations
were made
even to the
second. The
Hindus
further say
that all the
planets were
under the
same zodiac
sign at the
time of
Kaliyug and
their tables
also say
so.
In his book,
Cosmos,
famous
Cosmologist
of Europe,
Carl Segan,
said, The
Hinduism is
the only one
which is
dedicated to
the belief
that a
particular
sequence of
the creation
and
destruction
of the
universe is
going on and
this is the
only
religion
which has
made
calculations
from the
ordinary day
and night to
Brahmas day
and night of
8 arab 64
crore years
which, by
coincidence,
is close to
the modern
astronomical
calculations.
This
calculation
is older
than the age
of the earth
and the sun.
Besides,
they possess
measures for
even larger
calculations.
Astronomy
Astronomy
was called
the eyes of
the Vedas
because the
behaviour of
all
creations is
determined
by time and
time is
known
through the
movement of
the planets.
The history
of direct
observation
and
accurate/exact
calculations
of time and
planets is
more than
6000 years
old. There
was an
observatory
in
Pataliputra
where
Aryabhatta
drew many
conclusions
after
observing
the sky.
Famous
astronomer
John
Playfair
(1790) says:
India had
knowledge in
astronomy
for more
than 6000
years and
that Indian
calculations
were applied
throughout
the world.
He goes on,
Hence, we
can say that
in 4300 BC,
astronomy
was
developed in
India on the
basis of
direct
experiences.
Architecture
Mohanjodaro
(Sind),
Dwarka,
Lothal Port
(Saurashtra),
Varanasi are
the living
legends of
Indian
architecture.
So about the
Lingaraj
temple
(Orissa),
Khajuraho,
Ramaeswaram
and
Meenakshi
temples. The
ancient
architecture
was quite
comprehensive
encompassing
in itself
the concepts
of town
planning,
buildings,
temples,
sculpting,
fine arts
and
literally
everything.
In Bhrigu
Samhita sage
Bhrigu says
that before
buying land
it must be
tested in
five
different
ways and
each item to
be used in
construction
should be
tested.
Vishwakarma
Vastushastra
and Kashyap
Shilp are
other
treatise on
construction
and town
planning.
Kanjivaram
is an
excellent
example of
city
planning.
India had a
great
history of
ancient
chemistry.
We had a
plenty of
Indian
chemists in
the persons
of Nagarjuna,
Vagbhatta,
Govindacharya,
Yashodhar,
Ramchandra
and Somdev
etc. who
wrote a
number of
authoritative
books based
on personal
knowledge
and
experimentation.
Nagarjuna
explained
how to
purify
mercury.
Until the
17th
century,
Europe did
not know
what mercury
was, but
India knew
it for
thousands of
years.
Shri Soni
concludes
quoting Sir
Jagdish
Chandra Baso,
the great
scientist of
modern India
who in
foreword to
one of his
books wrote:
To my
countrymen
Who will yet
claim
The
intellectual
heritage
Of their
ancestors.
The book
contains
chapter-wise
references
and is
profusely
illustrated
with
diagrams,
illustrations
and sketches
to make the
reader
understand
the subject
easily. The
book is well
produced
with an
excellent
get-up. The
avid reader
will
definitely
derive a
sense of
pride and
pleasure by
discovering
the hitherto
facts
unknown to
the common
Indian. The
effort of
the author
needs to be
applauded.
It is a book
every Indian
should read.
It will
instil in
him a sense
of pride in
the country
and inspire
him to
achieve
still
higher.
Another
important
point that
emerges
after a
study of
this book is
that every
student of
science who
wishes to
achieve
something
must acquire
a working
knowledge of
Sanskrit and
also study
our Vedas,
Puranas,
Smritis and
other
classics
which can
only enhance
his
knowledge,
sharpen his
intellect
and motivate
him further
lest Sir
William
Jones has to
say that
Had Linius
(the father
of modern
science of
classifcation)
learnt
Sanskrit, he
would have
developed
his method
of giving
names more
perfect.
(Ocean
Books Pvt.
Ltd; 4/19,
Asaf Ali
Road, New
Delhi-110
002.)
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Links
Chronological list of Mathematicians
Indic mathematicians through the ages
FAQ On the Mathematics
of the Vedics
The Indic Mathematical Tradition
Mathematics Historians
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. Ockhams 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 civilization
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
Was calculus invented in India ?
Guidelines for
Publication
Decimal vs Binary The advent of the computer has given rise to
number systems other than Decimal, for example Binary ( base2 )
and hexadecimal systems (base 16). It can be seen that as the
value of the 'base' decreases the more the number of
digits that are needed to represent the number. This is of no
consequence for a computer but for the ancients who had to
indulge in a lot of mental arithmetic the base number had to be
as large as possible while remembering the properties of the
first (base -1) numbers. Obviously the choice of a decimal had
something to do also with the fact that we have 10 fingers
and 10 toes.

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