Who are We?
What do we do?
Latest News
Free Resources
Links
Professional brochureprinter 
FAQ in Ancient
Indian Mathematics

What were the
strengths of the ancient Indian Mathematicians
The Indian
Mathematicians of the ancient era were primarily number
theorists. Their interest in this field arose from a need to do
astronomical calculations, which in turn were needed to advise
the farmers on the proper timing to plant crops and for
performing rituals . In particular they excelled in Diophantine
Equations, algebraic equations in which only solutions in
integers are permitted. Examples of Diophantine equations
that the ancients dealt with are

ax
+ by = 1: this
is a linear Diophantine.

x^{n}
+ y^{n} = z^{n}:
For n = 2 there
are infinitely many solutions (x,y,z),
the
Pythagorean triples. For larger values
of n,
Fermat's last theorem states that no
positive integer solutions x,
y, z
satisfying the
above equation exist.

x^{2}
 n y^{2} = 1:
(Pell's
equation) which is named, mistakenly,
after the English mathematician
John Pell. It was studied by
Brahmagupta
and much later by
Fermat.
Another field or
approach which the ancients favored was algebraic geometry,
where geometry was studied predominantly using algebraic
equations. Again the need was for designing ritual altars and
their proper orientation with respect to the heavenly bodies and
for following the rules of Vastu sastra (architecture) .
Of course, the most
important single concept that they developed was the decimal
place value system. While other civilizations like the Maya and
the Oaxaca of ancient Mexico developed place value systems
independently, there is no evidence that they mastered the
technique of using numbers with such facility as the
Vedics did. The Vedics also developed facility with areas such
as Trigonometry and there is mention that they manipulated
arrays of numbers in the same manner as we use matrices today.
Even with all these
skills, it must be recognized that this forms only a small part
of what constitutes the corpus of modern mathematics today. It
must also be recognized that the Greeks and the Babylonians also
had developed mathematical prowess, if one accepts that the
axiomatic approach to geometry as primarily a Greek development.
Despite all these
caveats, it must be admitted that for the era in which
they lived the Vedic contributions to the sum of human knowledge
especially in mathematics was considerable and should therefore
be a matter of great inspiration for those of us who consider
themselves part of the Indic civilization. Further there is much
study yet to be done and many more manuscripts which remain
uninvestigated so the quest for what was the state of knowledge
of the ancients in fields such as mathematics has just begun

Why do you use the term Vedic
mathematics
Strictly speaking the later contributors
like
Varahamihira did not live in the Vedic era, but the methods
used show a continuity in development till the Modern age and
hence the use of the term Vedic , signifying techniques which
came into use in the distant past seems justified

Why even bother with Vedic mathematics
when we have progressed much farther in the intervening
millennia ?
I will quote the answer I gave in a
discussion forum (BharatRakshak)a few years ago. "
First, do we understand the corpus
of VM and Vedic Science (i certainly
dont know enough about VM to make an
authoritative statement). What part
of what we know to be contributions,
were purely Vedic and what part were
contributions by mathematicians like
Aryabhatta and Bhaskaracharya who
came much later ?
Second what part of it is already
accepted in modern mathematics ( and
I am not talking about arithmetical
tricks to do multiplication and
division). I remember in my Hall and
Stevens text on geometry which i
used in India, the proof of
Pythagoras theorem took up a whole
page. I would have loved to have the
4 alternate proofs offered by the
VMB. I am interested for example in
the development of Astronomy.
Ptolemy used a concept called
ecliptics (if i recall the use of
epicycles  pl. dont harangue me on
this, as this is of the top of my
head)to get around the fact that the
earth went around the sun and not
the other way around. Did he borrow
that from Vedic or was that a
different stream of thought (Ptolemy
of course predates alKhwarismi and
his 'Sindhind zij'). Given that
Ptolemy post dates Vedic why did alKhwarismi
choose to rely more on Vedic ( a
more ancient technology) when he
presumably had access to both ?
Third the point made by James is
valid. The point is not to judge
Vedic Science by the standards of
21st century and then trash it
saying it is outmoded. Of course it
is outmoded, in certain respects,
you should expect that after 6
millenia. If it was not, that is
tantamount to admitting that we have
made no progress in the intervening
millenia. The point is, does it give
an alternate model at looking at
nature (e.g. ayurveda) that is
equally valid, useful and perhaps
more elegant.
Fourth, these efforts at
understanding our past, should not
be restricted to Vedic systems but
to other systems in India's past
such as Yunani (etymology ionian =
Greek) which is attributed to
Islamic savants. I am interested for
example in tracing through the
development of algebra (coined by
alKhwarismi as algibr wal
maquaballah) from its ancient Vedic
origins because of the efforts of
Islamic savants in the middle ages,
at a time when Europe was in the
dark ages and was struggling with
Roman numerals.
My own view is that Vedic science
and math is a forgotten science and
for the most part does not
contradict what has been discovered
subsequently. First let us
understand what it says before
getting an anxiety neurosis that it
is going to replace Western science
in Indian schools.




Where are these results to be found in
the ancient texts
The earliest works on Mathematics by the
Vedic savants are recorded in the Sulvasutras, the sacred books
of altar construction in the Vedas, in particular the Apastambha
Sulvasutra, the Baudhayana Sulvasutra and the Katyayana
Sulvasutra. The Sulvasutras are appended to a particular Veda
(see FAQ on Hinduism
for the typical contents of a Veda) can be translated as rope
rules or "manuals of measurement", the modern term for which
would be metrology. But in reality the scope of the
investigations in the Sulvasutras is far broader and comprises
among other fields, number theory, trigonometrics, algebra,
algebraic geometry, series expansions, the concept of a
rational number, etc.etc. The dating of these Sulvasutras, while
occurring after the main corpus of the Veda was compiled,
is of great antiquity, greater than that of Babylonian
mathematics. It is only now that we are beginning to understand
the extent of the antiquity of these ancient mathematicians. See
for instance The origin of Mathematics by Lakshmikantham and
Leela. in the list of references later in this page.
A word needs to be said about the use of
Sutras (Aphorisms) as a means of communication and recording of
results. The dictionary says that a sutra is Any of various
aphoristic doctrinal summaries produced for memorization
generally during the millenia before the common era and
later incorporated into Hindu literature..We
must recall that writing materials during that period of
history, were not plentiful and had to be laboriously produced
probably by the author of the sutras himself . Whatever needed
to be communicated had to be in as brief a form as possible.
Hence the need for economy in language and the use of sutras. In
fact there is evidence from which one can infer that the Vedics
were the first to use symbols and mathematical equations and
hence their prowess with Algebra . Brevity has its downside
however and the charge has been made by Europeans that the
Vedics rarely provided proof. The reason was that the proof was
generally terse and incomprehensible to the majority of the
people. The notion that those who mastered these topics used
brevity as a means to restricting this knowledge, overlooks the
fact few in the populace would have the capability to pursue
this rigorously without the intellectual discipline that
comes from years of study. Then as now Mathematics had the
reputation of being a difficult subject to master.

Who were the main contributors to
Vedic Mathematics

Why are the contributions of the Vedic
savants as widely known as those of Greek, Arab
mathematicians
In the early years of colonial rule by the
British (the attitude persists among Western Indologists even
today, although less so among mathematicians in the West)
there was great reluctance to believe the sacred texts, after
they were first recognized by Sir William Jones in the
1770's. A typical reaction was that of W.W Rouse Ball in
History of Mathematics . I posted the following in the Bharat
Rakshak forum in2000
"Typical
of the racism exhibited by the Brits and other Europeans is W.W.
Rouse Ball in 'A short account of the History of mathematics'
Dover Publications,1960, (originally appeared in 1908), page
146'The Arabs had considerable commerce with India, and a
knowledge of one or both of the two great Hindoo works on
algebra had been obtained in the Caliphate of AlMansur (754775
AD)though it was not until fifty or seventy years later that
they attracted much attention. The algebra and arithmetic of the
Arabs were largely founded on these treatises, and I therefore
devote this section to the consideration of Hindoo
mathematics.The Hindoos like the Chinese have pretended that
they are the most ancient people on the face of the earth, and
that to them all sciences owe their creation. But it is probable
that these pretensions have no foundation; and in fact no
science or useful art (except a rather fantastic architecture
and sculpture) can be definitely traced back to the inhabitants
of the Indian peninsula prior to the Aryan invasion. This seems
to have taken place at some time in the fifth century or in the
sixth century when a tribe of Aryans entered India by the north
west part of their country. Their descendants, wherever they
have kept their blood pure, may still be recognized by their
superiority over the races they originally conquered; but as is
the case with the modern Europeans, they found the climate
trying and gradually degenerated. Note the blatant racism in the
second paragraph and the venom that this author exhibits.[This
message has been edited by Kaushal (edited 15062000).]
"
Thus
there was great reluctance to admit that the dark skinned
natives of the Indian subcontinent could be capable of
intellectual effort. Even after the advent of the legendary
Srinivasa Ramanujam, the great number theorist in the
early years of the 20th century, from what is now Chennai,
Tamil Nadu , such attitudes among British and European scholars
were hard to dispel. We will have a lot more to say about
Srinivas Ramanujam later in these pages.
With
the coming of the internet, and the great proficiency of the
Indics in matters related to Information Technology, this state
of affairs has begun to change. Both the Indics and Western
savants have begun to realize the profound importance of these
early developments in mathematics to the advancement of human
civilization. See for instance a recent column on
Place Value systems.

References on Vedic mathematics

Links on Vedic mathematics
ŠKosla Vepa,2006 
