Actually, this subject matter
that I have to discuss today
has become the center of controversy
in recent times.
Because you have heard that
India's eminent personalities have been
heard saying that Ganesha's head
was transplanted because at that time
in that very ancient period,
surgery had developed to such an extent.
We heard that Karna's birth was because
of the development of medical sciences
at that time.
We heard that
some 7,000 years back,
our ancient sages used to fly in aircraft
that went not only to different places
but also planets.
We have heard all that.
And that has naturally
raised a question.
What has happened is that
there are some people
who believe all that,
and there are some people who have the
opinion that practically nothing happened
in India.
And that science developed
when the Western Science came.
What I will try to show is that
both these notions are false.
India did have
very rich scientific heritage.
But we need to understand
what exactly that is.
And in giving this talk, I will
make a request to the whole audience.
The request is this.
Never allow a speaker to say anything
without proof. Including me...
So, whenever I say something,
it is within your rights to demand proof
And it is my job to say only
the things which can be proved.
So I'll say only that.
In discussing the history of science
in India, what I'll do is
I will divide the whole period
of ancient India into a few epochs.
Normally, we do that in studying history
certain period is called medieval period
certain period is called ancient period
There is a reason for that.
Without that
you cannot organize thought.
The time-periods in which
I will break it are
(1) Indus Valley Civilization,
(2) The Vedic Age,
(3) The Post-Vedic period,
(4) The Medieval age
and
(5) the Modern age.
And the time-periods are
more or less displayed here.
The moment I do that,
you might ask me
"What is the proof of these times?"
Well, whenever historians
want to find out times,
they depend on 2 types of things.
One is called Monuments,
second is called Documents.
Monuments are Things that are found
and which, with radiometric dating,
can be dated.
And two,
Documents.
If the documents talk about
the reign of a particular king,
we will know that
that document's period
and his period are same.
So, there are very identifiable methods
by which
scientists or historians
identify these ages.
So for example,
Indus Valley Civilization have those
relics from which you can date.
Unfortunately, in the Vedic period,
we have nothing. Not a single monument.
Monument also includes houses,
remains of the houses,
remains of the things
that they used -
if they fought, then some bows or
arrows would be found... something.
But we have little.
So let us start with
the Indus Valley Civilization.
The Indus Valley Civilization
essentially
is concentrated in the
North-West side of India, and Pakistan.
We hear about Mohenjodaro and Harappa
but actually there are more than 500 sites.
It covers around this much...
the most important thing is that it is
an urban-civilization, not village-based
Since we are discussing not the history,
but science ...
so I will point
specifically to that.
This is a mound.
And close to that, at some elevation,
we have a granary.
And here we have a public bath
and we have bigger structures.
You also do see these kind of drains.
And from the houses, there are
drains connecting to the public drains
and those go all the way to the rivers.
You have these wells and we also
have rectangular canals off the houses.
These are bricks, burned clay
bricks of specific sizes, 1:2:4,
That is true for whole
Harappan civilization.
One important aspect is that there is a
regularity of the plan and construction.
That can only happen if
there is some kind of Government.
That cannot happen if
everybody builds on their own.
Another important point is that
there is no monument, no big house,
which means that no empire or king
Now let us try to figure out what kind of
science they needed in order to make this
Suppose you are given the task
of going to a barren field
and drawing a rectangle.
You will immediately realize that
it requires some idea of geometry,
especially, how to draw a rectangle,
draw a right angle,
how to draw equal sides, all that will be
needed. That much was known at that time
And we see a system of
weights and measures.
There was a scale and they had
weights and weighing balance.
And, in their Terra-cota seals,
you find various types of designs.
In order to make designs like these,
you need to have very clear (?).
Without that,
you cannot make these designs.
So that much of (?)
Why you cannot (?) master...
because
their script ...
They had a script but they (?)
This civilization went through a period
of decline and finally extinction.
And we have not been able to
figure out exactly why.
But the point is that such ancient
civilizations in various places
had also seen such decline
and disappearance.
For example, in South America,there were
3 civilizations- Aztec, Maya and Inca.
When Christopher Columbus went,
which civilization did he encounter?
It was Aztec.
What happened to Maya and Inca?
They disappeared.
So, the relics are there.
which means that there was a civilization
that existed for 500 years or 600 years.
The reason most scientists (?)
In order for such city-based
urban civilizations to develop,
since there is concentration
of human population,
there has to be supporting area of
agriculture that feeds
that concentrated human population.
If somehow the agriculture declines,
then that becomes unviable.
That's one possibility.
The other is diseases.
If you have concentration of people,
infectious disease can spread very fast.
And that can also
eradicate a civilization.
Any of these may have happened
but we do not know what exactly happened
But we do know that after around
this time, it declined and disappeared.
How do we know this time?
Radiometric dating.
Nothing is found dating after this.
So, with that, let us go the Vedic age.
As I say, the Vedic age,
we have various records
mainly of the nature of documents.
The currently established idea is that
a specific language group migrated
from Asia minor to various places.
And one of those tribes or many of
these tribes entered around this time.
If you want proof of that,
I will come to that later.
These were horse-riding, pastoral,
migratory tribes
and they did not know agriculture
but their main livelihood
was animal husbandry.
And they did not have
a written language.
And because of that...
you know that sruti and smriti...
that is how they had to
propagate their literature.
This is the picture
of the migratory routes.
And out of that, one route
leads to India.
but all these places are populated
by the people who at one time
migrated from here.
The main literature that we have
from that period are
Vedas, Vedanga, Vedanta, Puranas,
Upanishads, Brahmanas, Samhitas.
These are intellectual creations
of the time.
And that is why, this is presented to us
as (?).
And naturally, when we talk about
the scientific developments,
we have to count on these literature.
Numbers.
We know that they used to write numbers
in a 10-base system.
But it was not the kind
of numbers that we use.
Why?
Because we use zeroes.
They had no 'zero'
We use the place-value system.
They had no place value system.
Yet it was a n-base system.
Which means that 1,2,3,4,5,6,7,8,9-
nine different symbols for nine digits-
and then for 10,
you need a new symbol.
After that, you place the symbol for ten
beside nine symbols to get 11,12,13,14,15,16,..
And for 20,
you need a new symbol.
Which means that for 20,30,40,50,60...
you need new symbols
For 100,200,300,400,...
you need new symbols, 1000, 2000...
for all these you need new symbols.
Yet you will find that they could
figure out very very large numbers.
Even though that means
they had to memorize
very very large number of symbols.
Zero had not been discovered.
And they needed umm...
Still, they could figure out
numbers as big as a trillion.
They could figure out what are odd numbers,
multiples of 4, fractions, such as 1/4
those things are
mentioned in various literature
And I'll come to the ...
in more detail
That's where the main development was.
It happened around this time
in what are known as Sulbas.
So there are 7 Sulba-sutra texts.
Out of these,
Baudhayana's sulbasutra is the oldest
and also it is most extensive.
So I'll mainly concentrate on the things
that are found in Baudhayana Sulbasutra.
What is Sulba-sutra?
You see, these are the ways
to make the altar ('Vedi').
And in order to make the Vedi,
they used to take a rope.
Sulba means a rope.
So, how to make the Vedi with a rope?
And since the Vedi construction
demanded certain geometry,
they had to tackle the problem
of doing that geometry with just a rope.
And that is enough.
If I ask you to tell me what is the
North-South direction here,
will you be able to tell me
without the aid of any compass?
They figured out that.
Their demonstration was like this.
You make a circle
by extending the rope around.
Put a pole here and observe the shadow
of the pole when the sun rises. Save this
Then observe the shadow
of the pole when the Sun sets. Save this.
Mark where it intercepts. And then
draw a line between the two.
This is the East-West line.
For example, how to draw a
perpendicular bisector with a thread?
Take a thread.
And mark the middle-point of the thread.
Take two points
smaller than the thread's length.
Now stretch the thread to this direction
And wherever this mid-point goes,
that will be your perpendicular bisector
So, nice methods of...
they were using.
And remarkably, even a statement
of the Pythagoras theorem
appears in this SulbhaSutra.
It says (in English),
"A Rope stretched along the length
of the diagonal produces an area
which the vertical and horizontal sides
make together."
It means that at least the statement
of the Pythagoras theorem was found then
This question was asked
to Professor Manjul Bhargava.
He says that (the question),
"Who invented Pythagoras theorem?"
The answer to that question depends on
what you are actually trying to find out
The first to have some idea
about it was Egypt.
Because you cannot really make those
Pyramids without any idea of geometry.
The first to create
a Pythagorean triplet ...
these are the triplets:
3-squared PLUS 4-squared
is equal to 5-squared.
5-squared PLUS 12-squared is equal to
13-squared.
These triplets
were mentioned in Mesopotamia.
So this is also another
statement of Pythagoras theorem.
And first to formulate the theoretical
statement is in
India (800 B.C. Baudhayana SulbhaSutra).
And the first to give a proof
was in China.
None of these are Pythagoras.
So, Pythagoras really learned it and wrote
it and we don't have any writing of it.
We learned 'this' is Pythagoras theorem
from whom?
Euclid.
Euclid's writing said that this is
Pythagoras theorem.
So we call it Pythagoras theorem.
But it was actually lot more ancient...
For example, there are some
approximations we find.
There is no decimal number.
So they had to do it like that.
For example, square-root of 2 was
computed to be something like that.
And we have another single sentence ...
this was not represented as an equation.
It was a full sentence.
But if you write it out,
square-root of 2 becomes this.
And it is really wonderful how close the
number is... true upto 5th decimal place
What was the need for it?
Because root-2 is very important.
Because 1-squared PLUS 1-squared is 2.
So the diagonal of a square of side 1,
is root-2.
For example, we find some theorems.
"Diagonals of a rectangle
bisect each other."
"Diagonals of a rhombus
bisect at right angles."
"Area of square formed by joining the
middle points of a square
is half the original."
so on and so forth.
This circling a square
is a very important problem
you find in all the Sulbha-sutras.
Why?
As I told you,
these are meant for making the altars.
And altars are for Pooja
(worship).
And puja is to Gods.
Now, there were various Gods.
Some Gods were bigger Gods,
some Gods were smaller Gods.
And some Gods had equal position.
If some Gods have equal position, we
need to make their altars of same size.
For somebody, we need to make an altar
of a square,
for somebody you need to make a circle,
for another a rectangle.
So we're given the problem of constructing
a square of the same area as a circle
or a circle of the
same size as a rectangle.
How to do that?
It's not an easy problem.
Most of you are students of an IIT?
If I give you the problem, can you
solve it and tell me?
You cannot.
It's not easy.
The way they did it was like this.
Circling a Square.
The square ABCD.
This is the mid-point O.
Draw OA.
Draw the circle centered at O and
radius OA (circumcircle).
Draw the line perpendicular to AB
from point O.
It intercepts the circle at point M
and the square at point N.
Divide this NM into ratio 2/3rd to 1/3rd
Then you will get the point P.
And draw a circle there.
That is, according to them, the circle
of the same area as the square ABCD.
Is it true?
Well, you can yourself do it.
And you will find the area of this circle
and then get the value of π (pi)
which comes to be around 3.08,
approximately.
We know that 3.14159 is the value of π .
So, they got something like 3.08.
So this construction gives approximate
value of π (pi).
As I told you, initially in the Vedic period,
there was no agriculture.
But later, around 1000 B.C.,
agriculture started.
And as in many civilizations,
they had to formulate a calendar.
And in order to formulate a calendar,
they had to look at the sky.
Because that's what offers you
some periodic event ...
They did that and with that,
they did observe
the motion of the Sun, moon and planets.
And that is what you find
in Vedanga Jyotishya.
By the way, do not believe that
Vedanga Jyotishya is jyotishyam.
No, it's not.
It deals with astronomy.
There is nothing like astrology
in Vedanga Jyotishya.
There was no astrology in Vedic period.
Period! None!
It came much later.
It came much later with/after Alexander.
So Astrology is not Indian.
They were conversant with the periodic
nature of Moon (amavasya and poornima),
the seasons
(they identified the 6 ritus)
& they could also identify what is known
as winter solstice and summer solstice.
Naturally, they had the idea about
the Uttarayana and Dakshinayana.
So, Uttarayana and Dakshinayana were
identified- clearly identified in calendar-
summer and winter solstices...
And they named 27 stars, the same names
we use still today in India.
27 stars along the path of the sun.
So, if you want to mention
where the Sun is or the moon is,
you can say that
'the Sun is in this Nakshatra.'
So that is how they needed it
and named 27 stars.
In medical science, we find that there was
a significant amount of development.
Even though some of those things
may not stand a test today.
But back then, it was ahead of the
other civilizations around the world.
And what Ayurveda achieved was that it is
basically collection of empirical knowledge
People tried  out various things & with that
certain empirical knowledge developed.
But what Ayurveda did was...
Ayurveda tried to systematize that knowledge.
That means there were all these small
knowledges (scattered) everywhere.
And Ayurveda systematized them.
There is a chapter on medicine for children,
There is a chapter on medicine for women,
There is a chapter on medicine for old age,
so on and so forth...
for cuts, bruises...
This kind of organization of medical
knowledge was achieved in Ayurveda.
One must appreciate the fact that
in any civilization,
anywhere in the world, there was
always a phase called the Magic phase.
What is the character of Magic phase?
You will find it is a very important aspect
anybody studying Anthropology should learn.
It is a very early period in human
civilization when people were
at the mercy of Nature.
Their life depended on Nature.
Whether or not they will be able
to produce anything depended on
whether they will be rain,
whether or not they will be
able to survive depends on
how many gifts are there,
son is there...
So their life depends on these.
If some are good, they wanted that.
If some are bad, they didn't want that.
And in every civilization, you will find
that these are seen in human image.
And the way you try to get something
out of somebody by pleasing them,
similarly, people tried various
things to please these.
These are human images.
So you will find that is a character
of the Vedic literature that
all that is they are essentially
trying to please with some offerings
and beverages
In anthropology, this phase of human
civilization is called the Magic phase
where we're trying to do various
things in order to please...
Please what? Natural phenomena.
This is important.
And the Rikvedic hymns
are essentially that,
attempts to please Natural things.
And that is why we find that there is a
healthy curiosity about Natural phenomena.
All these things they did because of
healthy curiosity about Nature.
But toward the end of the Vedic period,
we find that
these rituals became
all the important stuff
- "what to do?" and
"what not to do?"-
and stuff like that-
that sort of stripped human curiosity
and that is why
from around 6th century B.C.
people started looking for
various alternatives
and that's when
Jainism and Buddhism developed.
There is one common
misconception that is
that 'spiritualism is the characteristic
feature of the Vedic times.'
This is not true.
There were very important philosophical
truths which are not Vedic.
For example, Lokayata, Nyaya-Vaisheshika,
Samkhya, etc.
These things are not in Vedic tradition.
These are from outside Vedic traditions,
especially the Lokayat was definitely
from outside tradition.
And these were materialistic.
In fact, Lokayata did not believe in the
existence of an atma or soul,
refused to accept anything on the
basis of belief,
& theorized that everything in the world
is made of 4 constituents (chaturbhutas)
and things like that.
All these things may not agree with
what you might call modern science
but these were materialistic.
They thought that everything is
made of material.
Nowadays we find various claims
being made regarding the Vedic period.
So, let us examine them.
Now, some people have claimed
that there was no Aryan migration
into India.
They are claiming that Aryans were
indigenous people of India.
There are also claims that the Vedic
culture was 10,000 years old
(and this is a very conservative estimate
because in the last Indian Science Congress,
when somebody said that people used to
fly in aircraft 7000 years back...
that means obviously they started
some time before that).
So someone says 50,000 years old,
some say 1,00,000 years old.
All the science and technology
that modern era has seen,
were known to the Vedic sages.
And the Indus Valley Civilization was a
part of the Vedic age.
These are the claims being made.
So let us examine these arguments
on the basis of evidence.
When did humans become human?
When did the human race start;
when anatomically (equivalent?)
humans evolved around 1,00,000 years back
and culturally evolved...
(culturally evolved means somebody
who can speak and use tools)
around 30,000 years
to 40,000 years ago.
So you cannot have any civilization
before 30,000 years ago.
Not only that, in the early part after
humans became human was Paleolithic period
(Paleolithic means Old-Stone Age)...
That was when humans know basic culture.
It was the hunter-gatherer phase.
We are not talking about
the hunter-gatherer phase here, right?
Agriculture developed all over the world
around 10,000 years back.
So, we are not talking about a time
more than 10,000 years back.
So, let's talk sense.
Let us not talk in terms of belief.
Let's see.
Let's take each claim one-by-one.
The Indus Valley Civilization was a part
of the Vedic era.
Is this true?
There are a few evidences
on the basis of which
scientists say that
'No, this claim is not true.'
Firstly, the structures of Indus Valley
Civilization were made of bricks.
Not a single brick has been found
dating from the Vedic era.
Not a single brick all over India.
Because the Vedic Aryans lived in houses
that were built from wood, bamboo & reed.
The first brick has been found in a
temple in Bhitargaon
(now in Uttar Pradesh near Kanpur).
This is the temple.
Some of you may ask 'What was the
building material during Asoka's time?'
For example, the Sanchi Stupa ...
there is obviously no brick.
This is made of wood (timber) and stone.
So, even in Asoka's time,
there was no brick.
Second important evidence.
There is no horse in Indus Valley.
All the seals that we have found,
none of them is a horse.
Different animals are there,
but no horse.
Third, all over the Vedic literature
we find horse.
There are in Rigveda itself,
more than 560 mentions of horse.
Obviously, there was horse
in the Vedic period.
But there was no horse
in the Indus Valley.
The Indus Valley people
had a written script.
We have not been able to read it yet.
But the linguists have at least been
able to say that
this was written from the right to left.
But there was no written script
in the Vedic age,
at least the early Vedic age.
That's why Vedic people had
to convey using Sruti and Smriti.
There is no description of city-life
(urban life)
or brick-built houses, grid-pattern of
streets, drainage system,
in any of the Vedic literature.
So if those people encountered the
Indus Valley civilization,
obviously there would be some mention
of these? There's nothing.
And the pantheon (Gods and Goddesses)
are also different
as evidenced from the images
that we have found.
Some people believe that Shiva
was a non-Aryan God
(he is an Indian God alright)
but not from the Indus Valley.
There were non-Aryans in India
with whom the Aryan people (?).
But there is no evidence of Shiva
in the Indus Valley Civilization.
There's another claim.
So, with all the 5 claims, we can
say that there was no cultural exchange
between the people of Indus valley
and people of Vedic period.
Had there been, at
least these things would have happened.
Had it been, the Vedic people would
have some kind of written statement
There was definitely no cultural
exchange between the two.
There is a claim that Vedic Aryans
were indigenous people of India.
That they didn't come from outside.
Main important evidence (refuting this)
is the horse.
The horse is the prime animal
in all Vedic literature.
Yet, horse is not found in
any of the Indian forests.
Horse is not an Indian animal.
Have you seen a horse going
around in Indian forests?
It is not an Indian animal.
So, before a certain time,
we have no horses
and after a certain time, we started
finding horse populations
means not that some horses were going
around eating grass and they came to India.
No, not like that.
Horse came because
people came on horseback.
So, there must be a certain time
in which those people came.
And the horse has been present in
South Asia from around 2nd millenium B.C.
(most likely).
The earliest uncontroversial evidence
of horse is around 1600 B.C.
Evidence 2 comes from linguistics.
The languages spoken in Europe
and in India have a common root.
And these languages are classified
under one group
called
the Indo-European family of languages.
There are various words you find in
common between our language
and the European languages.
For example,
brother in English, Greek phrater,
Latin frater, Sanskrit bhrata.
Father in Greek is pater
(that's why we say paternal uncle),
Latin it's pater and pita in Sanskrit.
So, you will find that large number
of words are in common.
(?)
And in Grammar
there are major similarities.
Case -
you have in Indian languages.
That is there in many of
the European languages.
Gender.
Masculine, Feminine and Neuter genders.
We have this in Indian languages
as well as most European languages.
The forms change depending on gender -
"main khaunga" , "main khaungi".
So, depending on the gender, the word
changes. "Aap jayenge?" "Aap Jaayengi?"
Same thing happens in German language.
In Indian languages, to address somebody,
we have "aap", "tum", "tu" .
And the verb changes depending on that.
"Aap jayenge?" "Tum jaoge?" "Tu jayega?"
(Sie gehen ...)
So the number, in Sanskrit you will find
singular, dual and plural.
Most of the languages are similar.
But this has survived in some of
the languages in Europe.
That is why people say that these
languages have a common root.
And linguists have even worked out
the lineage of the languages.
The lineage of the languages is very
important ... there's another issue.
Languages allow us to date.
Every year, Oxford dictionary employs a
hundred people to find out what new words
have been added to the English language.
And every year, the dictonary becomes
a new one. Why?
Because the language is changing.
If you find any literature that talks
in terms of 'giveth', 'cometh' -
will you say it's recent?
No.
You will say it is at least 100 years old,
right?
By the way of language use...
... from that you can figure out the time.
And this is how people... linguists,
specialists in language,
can figure out the timings of divergence.
Any way... So, the linguists
have now figured out
that the original language family
branched out into 2 language families.
One is the Central languages (satem),
another is centum languages.
The centum languages created all the
European family (the one that went West).
And you can see the modern languages last.
These are called Intermediate languages
most of whom are extinct now.
Like if (?) that evolves and finally (?)
survived ... (?)
The Proto-Indo-European is sometimes called
the Proto-Indo-Iranian branch of languages.
And here at the end, you find the
current languages.
& these have started from one common root
& that's called the origin of language.
That is why most historians believe that
a few waves of migration of Aryan-speaking
people occurred in the period
from 2000 B.C to 1500 B.C.
But there is no evidence of warfare in
the Indus Valley cities.
There is no plundering, burning...
there is no evidence of that.
So it is believed that there was no
direct invasion into the cities ...
Some people earlier believed that
Indus Valley Civilization disappeared
before the Aryan invasion.
No.
Another important thing is that Aryan
is not a race.
It is just a language group.
It has never been categorized as a race.
Calling it a race was the ceration
of the German fascists.
Before the German fascists nobody
called it (Aryan) a race.
But after German fascism, many people
have started calling it a race.
It's not right.
It is just a language group.
So let us now come to...
if I have been able to convince you
that the Aryans were really people who
at some point of time came from outside,
they brought with them their language
but settled here.
And they had extremely rich culture,
extremely rich literature
from which we can still learn a lot about
ourselves.
Now let us come to the Post-Vedic
(Siddhantic) period.
Almost all the things that really are
startling come from this period.
When does this period start?
You can say around 6th century B.C.
This is the time when Jainism started.
6th century B.C. is when Buddhism started.
and after that it was a (?) period.
(?)
Though the language of the time
was still Sanskrit,
anything written in Sanskrit should not
be treated as Vedic literature.
No.
Even 200 years back, in India,
most of the literature that we have
were written in Sanskrit.
In medical science, we find great names:
Sushruta, Athreya, Jivaka, Charaka.
Jivaka was a physician.
But nevertheless, we know that they
developed surgery to some extent.
And what surgery they developed, that is
also there because these books
- CharakaSamhita and SushrutaSamhita -
have been found.
We have found them so we know
exactly what they are.
We found that they had discussed all
these surgical instruments.
It is real.
That development was there.
& there is a reason why in CharakaSamhita
as well as SushrutaSamhita,
their time-period is not very clear.
In SushrutaSamhita, the advantage is that
he mentions the king of that time.
So you can find out that this was
6th century B.C.
So at that time,
these developments were there.
This Sanskrit was different.
Vedic Sanskrit was not standardized.
That is why it is difficult to read
Vedic Sanskrit.
Because standardization means well-set
rules of Grammar with which
anybody can learn and
understand something.
That was not in Vedic sanskrit.
Panini did that.
That is why he is called the greatest
Grammarian of the ancient period.
And not only that, there is one thing I don't
know whether you have noticed.
That our alphabets have
a very curious structure.
For example, here the consonants are
arranged depending on
how much the mouth stretches.
This is based on how much air is exhaled
while pronouncing an alphabet.
And this is where the tongue touches
the roof of the mouth.
So, depending on certain complexities,
it is extremely scientific.
It's not there in any of the other languages
anywhere in the world.
It is India's contribution for language.
In Astronomy, for example, Siddhanta
Jyotisha was after the Vedic period
into the Siddhantic period.
And it was a collection of quite
a large number of books.
Do you notice that among these books,
you suddenly find Romaka Siddhanta,
Poulish Siddhanta, Yavana Siddhanta.
You can see that these are
Roman, Polish, Greek influences.
So from there astronomical or rather
astrological ideas entered India.
After that we find astrology in India.
Before that there was none.
But important thing is that at that time
they did observe the sky somewhat intensely
and they noted the Sidereal year,
they noticed this time-period.
But important thing is that they didn't
have the decimal point.
(?)
You notice the problem they faced?
How long is a day?
The year is these many days.
Even if they figured it out...
how do you figure it out?
The sun rises at different places on
different days.
After a certain period, it again rises
in the same place.
That is the year.
But they noticed that it's not exactly
the same place.
There is an offset.
And that offset... they found it and
so after how many of the revolutions will
the Sun be seen in the same exact place again?
You see the problem. And that was an
extremely large number of years.
And they pin-pointed that. This is the
number of years - 4,320,000 years.
This is called Mahayuga.
What is the Mahayuga?
It's essentially the number of years that
goes (pass by) before it becomes periodic.
It apparently is not a periodic function.
After how long does it become periodic,
that was the question they asked.
Now let us come to some of the great
people of that period.
As I say, there are some
very important personalities.
Aryabhatta is one. Since we have his book
Aryabhatiya, we know exactly what he said.
And there are some misconceptions about
that which I will speak about later.
He was the first who hypothesized that
Earth is rotating.
Now this is not the same
as modern theory of Solar System.
What he says is that he
'can see the Sun going from one place
to another,
& the moon going from one place to another,
or the stars going from
one place to the other.
All these are moving... it means that
we are moving. Earth is moving.
That was revolutionary for that period
so much so that
apart from him himself,
no one believed it.
Astronomers of his time - Brahmagupta,
Varahamihira - criticized him for that.
So they didn't accept it.
But he wrote that. That's important.
But he did not reach the Copernican model
(sun-centric model).
His explanation of the motion of the
planets was similar to that of Ptolemy.
We have in Mathematics, a great leap
forward with Jain mathematicians
who did some work in
Mathematics in this period.
Their main contribution was...
before that in Vedic period,
mathematics was subservient to... what?
Why did the people who wrote the
SulbhaSutras do this mathematics?
Because they wanted to make Vedis (altars).
So, it was not mathematics independent
of something.
But Jain mathematicians were fascinated
with mathematics independent of all this
(religious & ritualistic reasons).
And they were the first who conceived
infinity; very large numbers first
and then infinity. Not only that, they
also talk about various types of infinity.
For example, we know that
the number of points on a line
is less than the number of points in a
square. Both are infinity.
But one is bigger infinity and
other is smaller infinity.
In this time,
there are many texts that are found.
And many important mathematicians.
Pingala was a very important mathematician
of this period in whose writings
we find the Pascal triangle,
Binomial coefficients, fibonacci numbers.
We find that in Pingala.
Pingala was around this time, 200 A.D.
And by the way,
Aryabhatta was later (500 A.D.)
Zero and the place value system
was discovered in this period.
When it was discovered,
we do not know that.
We see its use mainly from the
5th century A.D. (A.D. not B.C.).
In Aryabhatta, we find this.
So, the discovery that absence of something
is also a number is a massive development.
Unimaginable development.
If this development was not ...
if this concept was not there, nothing
would have been there in mathematics.
And this is an Indian contribution.
The use of zero in writing numbers
(for example 20,30,40)
you don't need to
write new numbers (symbols).
People used to write separate symbols
for these in Vedic period.
The place value system of writing
bigger numbers like this
3905 =
3x10^3 + 9x10^2 + 0x10^1 + 5x10^0
All this was a development of this time.
And many people later,
scientists, acknowledged them.
Einstein, for example, wrote
'We owe a lot to Indians,
who taught us how to count,
without which no worthwhile scientific
discovery could have been made.'
An important evidence of that period ...
What I am saying
there is a basis for that.
The Bakshali Manuscript.
It was found in Pakistan,
in a very remote place.
This has been dated to around 400 A.D.
But there is a controversy over that.
This is a compendium of various rules &
illustrative examples and their solutions.
In this we find.... although it's torn &
we cannot read the whole thing...
but it is here that the
author is solving mathemtical problems.
And solutions are also contain algebra.
That's important.
So, let us look at the time-scale
of the Indian mathematicians.
Sulbasutras were in (800 B.C to 200 B.C),
Pingala was in 200 B.C.,
Jain mathematicians were in this time
(400 BC to 200 AD),
Bhakshali manuscript was in
4th century A.D.
and after that we have
these great developments.
Aryabhatta (5th century A.D.),
Varahamihira (6th century AD),
Brahmagupta (7th century A.D.),
Mahavira (9th century A.D.),
Bhaskaracharya (12th century A.D.).
After that we do not see any
big developments in the mainland,
the main part of India. Only in one corner,
Kerala, we find some development.
These are known as
the Kerala mathematicians.
And again, it was only with
Srinivasa Ramanujan in 19th century.
Aryabhatta, for example, in mathematics,
calculated π (pi) corretly
to 4 decimal places. (3.1416).
He formulated the first trigonometric table.
That was something. There was some...
For example,
you have an equation like this.
Two unknowns (x & y).
So it is indeterminate.
Yet they found solutions
with the assumption that
(?)
This is a very important problem
in number theory.
What whole numbers can satisfy
an equation like this?
And you will find solutions for this.
The Aryabhatta solutions of
Diophantine equation of this form
and ... this is a typical
(?) in Aryabhatiya.
And in Brahmagupta, we find cyclic
quadrilateral equations like this.
And this... is a second order
Diophantine equation.
And they found solutions.
And not only that,
Brahmagupta even formulated a method
of finding newer solutions for this.
So this is an outstanding contribution.
The square-root of negative number
doesn't exist.
Mahavira was first to have said that.
Much before Europeans.
And combinations of n objects taken r
at a time, what we say now nCr.
That was found in Mahavira.
Bhaskaracharya was the last of
the Great mathematicians.
And his contributions
are really outstanding.
He wrote on
arithmetic, algebra, permutations,
progressions, geometry,
mensuration and puzzles.
Puzzles were something he gave to his
daughter, Lilavati, to solve.
So, he taught his daughter by giving
puzzles and asking her to solve.
And finally he wrote a book
giving the solutions.
And the solutions are extremely beautiful.
For anybody to learn how it's done,
it's extremely interesting.
So it is clear that algebra developed here.
This is to be understood clearly.
Greece mainly developed geometry.
There was practically
no arithmetic and algebra in Greece
You know why? Because they
did not develop the number system.
And the whole thing depends
on the number system.
You cannot have good arithmetic...
Try to multiply two numbers in Roman ...
That is why in Greece, you will find
great developments in geometry.
Arithmetic and Algebra
mainly developed in India.
But not great geometry.
Not comparable to the way
the Greeks developed it;
like their method of axioms and proofs
and theorems, that we do not have.
Algebra definitely started in India.
And when Brahmagupta's work was
translated into Arabic
(from sanskrit to arabic),
and then it went into Arab world.
And we have great Arab mathematicians
who adopted algebra, developed it further,
(it's not that they only carried,
they also developed it further) ...
So the reason we call it Algebra because
that is the name Al-jabr.
He was an Arabian mathematician.
But he acknowledges this.
That it was from Brahmagupta
that I am taking.
Then he (Bhaskaracharya) also did a lot
of work in astronomy, naked-eye astronomy.
As far as naked-eye astronomy can go.
He was not an astronomer but he
essentially talked about how to observe.
And there is a book by Bhaskara
called Siddhanta Siromani
and it's one of the best
and exhaustive texts in Indian astronomy.
And he discovered moon's evection.
There is a chapter on spherical astronomy.
A different kind of geometry applies there.
And Bhaskara developed
that spherical astronomy.
There is a book called Goladhyaya that
actually talks about spherical astronomy.
And he had astronomical instruments...
& he improved the formulae and methods...
If you ever go to the Jantar Mantar (not
the one in Delhi but the one in Jaipur)
you will find some of these.
But you must know that Jaipur Jantar Mantar
was developed in 16th century after Galileo.
After Galileo means by then
already Europeans had started seeing
astronomical things with telescopes.
And then the Maharaja Jai Singh made it
as much as naked-eye astronomy.
It was this method
that was used in building it.
Not this method directly because Bhaskara's
work was translated into Arabic
and then re-translated back to Sanskrit
and only then used in building...
He introduced the concept of instantaneus
motion (tatkalika gati) of a planet...
Planets have average motion and specific
motion (?)...
Not only that he talks about sthula gati
(average velocity)
and sukshma gati
(accurate velocity).
And the trigonometric tables for example, ...
I did not discuss that.
In Brahmgupta, (we find a part of
trigonometry in Aryabhatta also)...
They used to call sine and cosine as
jyā and kojyā.
So they had a different name. For example,
in algebra we use x,y,z ...
They used kā (kālaka black), nī (nīlah blue)...
to represent x, y... variables etc.
They used the first letter of a color's name
blue, green, yellow...
For example, Blue , take 'b' out. That is
how they used to do algebra.
And you find various formulae, basically
the formulae that you had in school
they are all there in the writings of
Brahmagupta & Varahamihira & Bhaskara.
sine-squared PLUS cos-squared
is equal to 1.
That's there.
The solution of indeterminate
Diophantine equations...
Bhaskara II gave a general method
called Chakravala.
And people say this is one of
the basic contributions.
And he illustrated it for N=61 and N=67,
for example.
An equation like this ... Using
algorithmic procedure, he figured it.
Fermat says that ... later
European mathematicians followed the same.
This equation is actually attributed
to John Pell.
But it is actually something
that Bhaskara II talked on.
People have said that it should not
be called Pell's equation,
rather Brahmagupta's equation.
The comments by important mathematicians,
for example,
'Bhaskara's Chakravala method is beyond
all praise.
It is certainly the finest thing achieved in
the theory of numbers before Lagrange.'
Regarding Fermat's challenge,
Andre Weil remarks,
'What would have been Fermat's
astonishment if some missionary,
just back from India had told him that
his problem
has been successfully tackled there
by native mathematicians
almost five centuries earlier?'
This is something we should be proud of.
These are things we should be proud of.
Not something that is imaginary.
These are real.
The credit of constructing a
Cyclic-Quadrilateral with rational sides
goes to Brahmagupta (628 AD).
The only cyclic quadrilateral that was
known to western countries till 18th century
was with sides 39, 52, 60, 25 and it was
reffered to as Brahmagupta's quadrilateral.
The German mathematician
Ernst Kummer says
'Brahmagupta's simple method enables us to
construct any number of such quadrilaterals.'
So, this is a cyclic quadrilateral.
So, he gave a method for constructing
any cyclic quadrilateral.
From Lilavati...
this one is from Lilavati.
This is one of the constructions
of cyclic quadrilateral.
And then we come to the
Kerala mathematicians
who developed infinite series,
power series and trigonometry.
And also we have an inkling
of development of calculus.
I do not say that they developed calculus
that would be taking it too far.
But if somebody says that the
rate of change of something becomes zero,
then it will be a peak,
you know they had some idea about it.
The Decline and Fall of Science
in Ancient India
You have to understand that
after this ancient period,
there was a decline in science in India.
And the last scientist we find in India
was Bhaskara.
But first... a summary.
Discovery of zero & the positional-weight
number system
(2nd century AD to 5th century AD).
Advancement of medicine and surgery
was in this time.
So this period is called
the Siddhantic period.
There were great developments
in this period.
But then after this, nothing. Zero.
Why?
The person who tried to answer this question
was P.C. Ray (Prafulla Chandra Ray).
And then, after the Renaissance,
we can see a large number of people
working in science in India.
What PC Ray says is that there are 3
factors contributing to
the decline and fall of science in India.
(1) Introduction of casteism
in a rigid form,
(2) Introduction of strictures
(do's and dont's) of the  shastras
(3) Spread and intellectual dominance
of the vedanta philosophy
which, he identifies, taught people to see
the material world as illusion.
And because of that, he says,
Science died.
This is what he writes.
Essentially, he says that
if you have a system where, a society,
where the doers and thinkers interact
that is the fertile place
for development of science.
But when caste system developed,
the doers and thinkers became separate.
There was no contact between the two.
And when that happens maybe your
dexterity with the hand may develop
but you cannot develop
a Boyle or a Newton
'Her soil was rendered morally unfit for
the birth of Boyle, a Descartes or a Newton
and her very name was all but expunged
from the map of the scientific world.'
On Shastras, he (P.C.Ray) says
'The drift of Manu...'
See, his point is that
in order to learn surgery,
a student must cut a dead body
to understand what is there inside.
Without that you cannot learn surgery.
Susruta also said that.
But in the Shastras when touching a cadaver
became something that you cannot do
(do's and dont's) then obviously the
cultivation of surgery died.
P.C.Ray commented that surgery
became practiced only by the barbers.
In medieval India, (?)
done by any religion.
There was a barber who used to cut.
And then for Vedanta philosophy, he says...
'The Vedanta philosophy, as modified and
expanded by Samkara,
which teaches the unreality
of the material world,
is also to a large extent responsible
for bringing the study
of physical science into disrepute.'
So he says that if you look at the
material world as an illusion,
then obviously you will not have
interest to know about it. (?)
You have to understand,
you have to believe that what you are
seeing, what you are touching,
the things around you,
are real.
Only on that you can ask
questions and develop science.
So when there was an intellectual
dominance of this kind of philosophy,
according to P.C.Ray,
science died.
Again, we had this Indian Renaissance
started by Raja Rammohan Roy
and Ishwar Chandra Vidyasagar.
And then we see
a galaxy of scientists in India.
So, it is because a change of
intellectual climate
that led to a condition conducive
to the development of science.
So, naturally, we need to cultivate that
intellectual climate
that's conducive
to the development of science.
Thank you.
