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so far we had 3 lectures on aryabhatta so
in this fourth part phone i will be definitely
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concluding aryabhatiya and then we will move
on to bit of jaina mathematics in the last
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lecture on aryabhatta i was explaining to
you about the methods by which aryabhatta
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has derived sin table so he has suggested
and then we also problems so problems can
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be handled with and in this lecture i will
basically start with series in fact
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i touched up on the arithmetic progression
which has been built by aryabhatta so in fact
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there are few other problems we can skipping
on ganitapada so it because highlighting two
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more problems and then i will move on to the
last part of ganitapada which displaces the
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kuttaka problem so kuttaka is a very interesting
terminology which has been used to refer to
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a certain process so by this we keep on reducing
the numbers so is basically hitting on head
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so
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it is basically founding up pulverising so
things are brought down ok so that is the
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set of the origin of the term kuttaka so formulation
of the problem then i will give you an example
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of how it has been used in astronomy in fact
the very purpose of the kuttaka seems to have
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been to solve pattern problems in astronomy
we will see a single examples of bhaskara's
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commentary and then i will explain in great
detail in the couple of versus so i think
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through
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which aryabhatta has presented this kuttaka
already so then i have an example then you
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move on to jaina mathematic even i say jaina
mathematics so i refer to the earlier part
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of the jaina mathematics the earlier part
of jaina mathematics has not been in samaskritham
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so so it is difficult to decipher the text
themselves and text themselves of not available
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and some of the commentaries are taken as
the resources for deciphering what happening
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there in the jaina
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earlier jaina text so with this i will first
of all introduce an interesting verse of aryabhatta
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this verse is presenting a formula by which
will be able to find out the number of terms
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in arithmetic series given that you know the
sum of the series and first term and the common
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difference so the verse goes like this refers
to the number of terms in arithmetic series
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in fact in the previous verse so we had the
sum mention the sum of the series the
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formula for finding the sum mention and therefore
in this verse is not explicitly stated but
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since aryabhatta as i was mentioned earlier
is more or less composed in sutra style in
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sutra style so we have something call so whatever
is not present in this sutra and it is available
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in the previous sutra we borrow it so this
is the kind of thing which seems to have been
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done here and since you are speaking about
the sum in the previous verse so here we have
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to exam
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so is the common difference so
is the first term
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difference so 2a-d and then varga is square
is addition so this term 8sd in order 2a-d
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square so this is what then what is to be
done take the square root of that -2a so sum
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here of this so series as i said is the common
difference the common difference is that forms
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the divisor then the last term is 1 add 1
to that and then he says divided by 2 so this
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is basically
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gives the number of terms in an arithmetic
series provided you know sd and a the formulas
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an interesting thing to see why aryabhata
has specified this in this particular form
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it has been commented by nilakantha wherein
he has shown certain way of moving this result
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not the alzebra but by geometrical construction
so you can think of ahh in fact the series
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can be considered to be ready ready is step
by step so a a+b and so on so similar kind
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of construction
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can be used to actually shows this result
so this result as such looks complicated and
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the if you going to show this kind of course
in various ways so right hand side can be
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written in various ways and this particular
way in which has been written seems to be
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coming from the fact that they might have
had a certain way of looking at it and that
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has been shown by neelakanta so which i m
not discussing right now certain algebraic
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identities
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have also been presented by aryabhatta so
just sample which i want to give aryabhatta
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has also discussed rule of three of course
professor sriram so in his talks will be covering
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all that in great detail but this mean the
earliest text so i wanted to present more
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or less so on various issues which aryabhatta
has discussed suppose you know the product
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of 2 number and you also know the difference
so this is the kind of problem which aryabhatta
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has chosen here we
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will you be able to find out those two numbers
so there is a question if x-y is a and xy=b
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then how do you find xy so this is the relation
which has been given by aryabhatta in this
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verse he says 2 square which is 4 is multiplication
to design a product so suppose you know the
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product of number numbers and that product
he refers xy let us say b so so
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if you look at this expression so which gives
the value for x and y the first term is and
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then ok four times b so the 2 numbers which
is considered is difference is square so you
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have to add this and then you take the square
root half of it so this is trace out this
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is one of the algebraic identify present and
it useful in various context so this algebraic
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identities have also been presented in the
form of verse some like to we memorise
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what is the ahh value of x in a quadratic
equation 2-b so that is kind of thing so this
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essentially in the form of verses now i move
on to the kuttaka algorithm so this kattakada
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algorithm is primarily a method by which you
will be able to find out solution to the first
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order indeterminate equation ok the indeterminate
equation we have 2 variables in this and this
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first order we have just 2 variables and step
by step procedure has been
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delineated by aryabhatta in 2 versus for solving
this equation and why were they worried about
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solving this i will show you an example the
context in which they had to necessarily find
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a solution for this so this finally to solve
astronomical problems so kuttaka is referring
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to the process of pulverizing something so
basically a repeated operation so even hammering
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so something is a repeated operation and here
what we do is so given two numbers will keep
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on
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successively divided one by the other so this
can be thought of it is similar to the repeat
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in algorithm and by mutual division we will
keep on reducing the magnitude of those two
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number and at some stage will stop this and
it can be stopped at any stage and then you
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can go and construct what the two numbers
x and y for this algorithm also plays a very
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key role
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in solving which i will also be discussing
little later when we discuss about brahmagupta
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and it like to be touched up on this kuttakara
occurs in various contact so therefore bhaskara
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classification
if it is arising the context of dealing in
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the revolution number of planets we call it
this can arise in various occasions suppose
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you want
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to express some magnitude in terms of degree
is sum in some other magnitude so if you have
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to handle those two so again there will be
a kuttakara so is basically some number so
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so all that requires kuttakara method so to
base the is the problem in more concrete way
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so let us consider a number n so this is how
we can easily understood the different ways
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in which this kuttakara problem can even be
presented so we choose one particular way
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which is very
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close to the method which has been described
by aryabhatta suppose this number n and this
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number when divided by a it gives some value
and when divided b it give some other value
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and then plus reminder plus reminder so n=ax+r1
is also equal to b/r2 so we it can see that
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so this can be represent in this particular
form we choose this by-c+ax where c is the
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difference of the two reminders r1 and r2
so the problem is given a b and c so we have
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to find
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interior solutions for x and y so this is
called kuttakara is all about so this the
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this is indefinite equation because we have
only one equation we have two variables so
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therefore it is not suitable to find a solution
in all the cases in certain cases we can easily
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guess the solution but in most of the cases
it will not be trivial and we need a certain
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systematic procedure by which will be able
to solve this equation from new point of history
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this equation has been generally
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referred to as diophantine equation so which
is not quite correct so if where to look at
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from the historical viewpoint in fact the
diophantus also attempted a slightly different
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kind of a problem he was not even trying to
find integer solution to this problem so he
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was trying to find rational solutions and
rational solutions of all more easier than
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getting integer solutions for this problem
so for so will give you the example one of
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the examples
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which has been presented by bhaskara to give
you your flavour of the context in which this
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kuttakara problem arises in astronomy in fact
bhaskara says so all that he says so i am
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going to present you how kuttakara arises
this is a problem so is basically the number
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of revolution made by the sun multiplied by
which factor in fact is a very important problem
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in the sense that indian astronomer have presented
a very large period so this for you
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to have a much general picture i just wanted
to spend a couple of minutes period so you
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have a large period so you can call period
that you want to define late to define let
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us define this and this large period to see
that the various planets which are moving
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in various orbits so they complete some integral
number of revolutions so there will be some
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reminder here and some reminders there so
now you have to see to it that you will be
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able to get a second period
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by which all of them can make integral number
of revolution so it is in this context with
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kuttakara naturally arises ok and here ahh
bhaskara says is basically the number of so
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number of mean the number of sunrises that
take place in a given period see even sun
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start moving or moon starts moving so i started
particular point so it will make some n number
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of solutions and when it comes back so it
may not be the same time so can we see
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that that after making integral number of
revolutions can fix certain period by with
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this is also company integral number of solution
and something else also will do so that is
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kind of context in which these arises he says
meaning
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of this period so so this is the kind of problem
which bhaskara wants to attempt with reference
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to the number of revolutions made by the planet
so now we move on to the kuttakara problem
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message so and
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the solution which aryabhatta gives these
verses are not very easy ahh so and have follow
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it carefully because the kind of compounds
which aryabhatta has formed and the terminology
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which he is employed may not be that familiar
given to those who wore fairly conversation
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with sanskrit so
this is the first half of the algorithm which
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has been given by aryabhatta so then the latter
half completes the whole algorithm the term
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normally means so tip kind of a thing so here
this has been used in the sense of getting
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a reminder see suppose you think of representing
a certain number so by this magnitude so if
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you keep on dividing this there will be some
reminder which comes so this is the tip that
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means so means reminder if you go back and
then see this see n=ax+r1 and by+r2 so r1
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and r2 are greater than r2 so if you call
r1 as and r2 as fine so may you divide
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cutting is referring to the process of division
here so in fact bhaskara starts his commentary
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by saying in common language we call it where
aryabhata has used the term so the remainder
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is large so here so this is a compound so
that is how we need to understand so the number
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so which is being divided ok in sanskrit confirm
various kinds of compounds so rameshwara when
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you say so rama is eshwara is one way of saying
or
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you can say so that he says so here what understood
as removing the compound from which we are
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removing the part so which means it is dividend
so may you divide when you say divide divide
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by what so by the other number for which the
reminder is small so so the process is end
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there this is the initial state in which you
get the first question then you will get some
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reminder take the reminder and then you so
divide the number by which you divided before
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for the process is going to be repeated so
for you to see
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this is the kind of the thing suppose a is
b so you have to do this so this q is first
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portion and then so r1 is a reminder so once
again you divide b so you will have something
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r2 is a reminder if you keep on doing the
process so this is what is referred to the
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aryabhatta he says is reminder so is so the
previous reminder is going to become the dividend
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now and this reminder so very few divide will
become the
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dividend the next stage so you have to do
it continuously then when you reach a certain
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stage so this can be terminated and at that
stage so he says you had to do a certain process
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that process is so multiplication so means
multiplied by so that is so what is this so
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here use that word ahh to refer to a certain
number which you are going to guess by making
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use of your budhi that is why we are called
he is calling it as mathi so mathi
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gunam means a number which will guess so you
have to make a guess so that the product of
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this mathi and the reminder that you have
so minus or plus something will be the device
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that will exactly be multiple of something
else ok just keep it in mind as see algorithm
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clear but mathi is basically referring to
a number which you have to guess at a particular
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stage so you can stop the division at any
stage in fact so if you carefully analyse
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the algorithm there is
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good algorithm so mathi gunam multiplied by
mathi so this is called an optional number
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is remainder is difference of the remainder
so that is what is denoted as c c refers to
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r1-r2 all that he says is so you hgave get
mathi so in this mathi has to be multiply
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by some number and to that this basically
it could be either addition or subtraction
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ok you have to do either addition or subtraction
is referring to either of it that can understand
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only from the commentaries
this number multiply by what see you have
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to multiply this
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by some number so and then +/-c so
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then he also says when should you add c when
should you remove c is even is odd so when
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the number of portions that you have see depending
upon the ahh process so you will get a certain
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number of portions the number of portion is
odd or have to do something if it is even
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you had to and he says if it is even you have
to add and when it is you have to remove it
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so how
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do you know this where he says so aryabhatta
has worked out this is how it seems to be
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so it is a which has been handed down to it
okay traditionally they have discovered this
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to be the method and therefore we interpret
it that way so this is how i explain so i
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move on to the second part see in fact this
stage of the verse ahh we have to see the
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commentary because what is to be done all
this so that has been very expressively stated
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by aryabhatta in
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the verse and therefore ok means to be guest
so ok if you divide so you will get a set
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of portions so you have to arrange them in
a particular way this has not been set in
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the verse so that is why the commentator say
something has to be means you have to make
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a guess so so i will quickly tell you we can
keep this in mind and then you can understand
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the verse so we got a we get a series of division
so we got a series of
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portions so all that has to be done is you
have to arrange them one below the other say
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q1 q2 bla bla 2n let us say up to that so
one by one below the other you have this is
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generally only means a sequence ok so all
the portions on the sequence and then below
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00:26:39,010 --> 00:26:45,970
that you have to place this mathi and then
some other number yes so which is actually
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00:26:45,970 --> 00:26:53,409
the portion which is obtained by doing this
operation so i said you have to make a guess
201
00:26:53,409 --> 00:26:56,510
of mathi so mathi into
202
00:26:56,510 --> 00:27:02,210
what is to be done is the last reminder that
you get so this time denoting as this for
203
00:27:02,210 --> 00:27:09,320
instance so you have this division process
be noted here and the last reminder is let
204
00:27:09,320 --> 00:27:21,250
us say r2n+1 now r2n+1*mathi ok so this is
guest so the number will be guess so how should
205
00:27:21,250 --> 00:27:33,100
we guess if it guess in such a way that r2n+1xt+cis
divisible by r2 by the previous ok so that
206
00:27:33,100 --> 00:27:37,929
is something which has not been stated in
the so that particular mathi also has to be
207
00:27:37,929 --> 00:27:43,471
placed over start a one below the other how
to place you have to place the mathi button
208
00:27:43,471 --> 00:27:54,750
below and and then
+/-c if it is divisible by some other r2n
209
00:27:54,750 --> 00:28:06,350
and then that portion is what is referred
to as means after making use of mathi and
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00:28:06,350 --> 00:28:11,169
after doing this division whatever be the
portion so that is the ultimate thing that
211
00:28:11,169 --> 00:28:15,250
you find out in the process after that it
is a matter of multiplication and getting
212
00:28:15,250 --> 00:28:20,640
the value of x and y so as the process end
here and therefore
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00:28:20,640 --> 00:28:28,510
something later you have to place that also
below so that is how it is stick get all the
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00:28:28,510 --> 00:28:34,370
portion one together place mathi and place
the portion that we get by doing this process
215
00:28:34,370 --> 00:28:40,370
ok then what should be done so this is the
arrangement so till that is over now i am
216
00:28:40,370 --> 00:28:50,360
going to the second verse which is given by
aryabhatta so so having created this now so
217
00:28:50,360 --> 00:28:57,030
what is to be done so the number which is
there an alternate number has to be multiplied
218
00:28:57,030 --> 00:29:05,309
by the previous number means you multiply
the number which is about that and then below
219
00:29:05,309 --> 00:29:12,400
that you place the portion so we have to add
that also so to show you here if you have
220
00:29:12,400 --> 00:29:20,179
done this for this stage what to do if you
have to take a product of p and qn q2n this
221
00:29:20,179 --> 00:29:35,870
is multiply and then whatever is below you
have to add that so
222
00:29:35,870 --> 00:29:44,380
this is how he describes the algorithm more
or less over so he says so this has to be
223
00:29:44,380 --> 00:29:56,799
divided by v smaller reminder so in fact both
a and b are referred to as so you may think
224
00:29:56,799 --> 00:29:59,380
that ac for instance if you look at this so
225
00:29:59,380 --> 00:30:07,659
ae/b how can this b called in that both are
in the sense so the main problem if we consider
226
00:30:07,659 --> 00:30:12,950
see is the main problem so a is division for
this b is also devise you have to get x and
227
00:30:12,950 --> 00:30:23,960
y ok so therefore both are refer to as here
so
228
00:30:23,960 --> 00:30:34,850
so whatever comes to the reminder ok so this
must be multiplied by the you find that being
229
00:30:34,850 --> 00:30:44,980
used here as well as here one not get confused
so is also refers to number n where it
230
00:30:44,980 --> 00:30:49,799
later says so b becomes the divisor and a
becomes another dividend that is a different
231
00:30:49,799 --> 00:31:00,110
thing ok so so when you add to that then we
will be able to get x y so this is the algorithm
232
00:31:00,110 --> 00:31:06,780
it has been described in by aryabhatta so
i will leave this so this is just the way
233
00:31:06,780 --> 00:31:11,890
the process of division with has to be done
and all the portions are obtained so you have
234
00:31:11,890 --> 00:31:16,929
to leave the first portion so when you count
the number of portions see this is something
235
00:31:16,929 --> 00:31:25,500
which has to be kept in mind the prescription
is so this c has to be added so that the portion
236
00:31:25,500 --> 00:31:27,580
are even and c has to
237
00:31:27,580 --> 00:31:33,960
be subtract so the portion are r so in doing
this once should not coming the mistake of
238
00:31:33,960 --> 00:31:40,269
taking q also in top so it is only after this
so the number of portion have to be counted
239
00:31:40,269 --> 00:31:53,559
so this division of b should be left out so
this is how things are arranged and q1 q2
240
00:31:53,559 --> 00:32:02,309
qn-1 2n t is mathi and then this see now what
is to be done having created this so you have
241
00:32:02,309 --> 00:32:08,929
to multiply that by this so qn*q+s so let
us call this is beta 1 and then the rest have
242
00:32:08,929 --> 00:32:09,929
to be
243
00:32:09,929 --> 00:32:16,740
this place like this when we move on to next
stage so this will be multiplied by this and
244
00:32:16,740 --> 00:32:26,990
then t will be added and then this will go
on c so qn-1*beta1 this quantity+t so this
245
00:32:26,990 --> 00:32:33,600
is algorithm and then when reach the state
where you have only 2 there is nothing more
246
00:32:33,600 --> 00:32:40,470
to be added you understand so at this stage
you stop this multiplication and creating
247
00:32:40,470 --> 00:32:51,600
the table so this is a serious of police finally
he says if you look at the verse so once you
248
00:32:51,600 --> 00:32:59,940
do that
means so
249
00:32:59,940 --> 00:33:09,070
this has to be divided by b b was referred
to as so a was so divide by that so the reminder
250
00:33:09,070 --> 00:33:17,210
whatever you get that is basically x ok so
we wanted to find out x and y so he described
251
00:33:17,210 --> 00:33:22,390
certain process and this is after you reach
the store numbers so then you make the product
252
00:33:22,390 --> 00:33:28,120
of this the last number with b and when you
divide that so whatever reminder that you
253
00:33:28,120 --> 00:33:40,289
get is basically yx ok so n=ax+r1 so let us
take an example so let us
254
00:33:40,289 --> 00:33:56,570
have this equation 45x+7=29y so what do we
do so we start this division process so 45/29
255
00:33:56,570 --> 00:34:06,130
what you get 1 and then remainder is 16 so
you take this as the dividend now so divide
256
00:34:06,130 --> 00:34:16,370
so you get 1 here 15 16 remainder is 13 and
so the 16 become divider now so you have 1
257
00:34:16,370 --> 00:34:26,860
so we have 3 as reminder so then 13 becomes
dividend so you have 4 as portion fine so
258
00:34:26,860 --> 00:34:34,530
how many portions we have in this it is odd
or even we have 1 2 1 3 so this is the case
259
00:34:34,530 --> 00:34:35,530
of having odd
260
00:34:35,530 --> 00:34:41,409
number of portions so what are the prescription
so prescription is when you have odd number
261
00:34:41,409 --> 00:34:51,450
of portion the c factor has to be subtracted
before that we need to make a guess of t so
262
00:34:51,450 --> 00:35:00,770
this this number so now we have arrived at
the situation where we can found so forming
263
00:35:00,770 --> 00:35:06,970
i just give q1 and q2 so on i have to make
a guess of this mathi now chine -7 to be divisible
264
00:35:06,970 --> 00:35:08,880
by the previous reminder northeastern project
question precaution recognize 12063 so these
265
00:35:08,880 --> 00:35:19,290
remainder last remainder is 1 so one time
t-7 should be divisible by the previous remainder
266
00:35:19,290 --> 00:35:30,950
so it is 30 then you had this so t is 10 for
that so i just put 10 so when i divide by
267
00:35:30,950 --> 00:35:39,270
3 the portion i get is 1 so because it is
3 so you just have 1 as the last so this is
268
00:35:39,270 --> 00:35:48,800
how it has to be arranged so at this stage
to having found this it is just a matter of
269
00:35:48,800 --> 00:35:59,040
simple multiplication and then creating for
the so 10x4+1 so that will be this number
270
00:35:59,040 --> 00:36:03,109
so this has to be taken as it is then 41x1+10
271
00:36:03,109 --> 00:36:11,109
that will be this number so 41 has to be taken
as it and then 51 will be taken as it is so
272
00:36:11,109 --> 00:36:21,580
51x1+ 41 is 92 over so all the value of so
finally having reach this stage what is to
273
00:36:21,580 --> 00:36:38,190
be done take the last number 92 and then it
was said if you look at the verse so he says
274
00:36:38,190 --> 00:36:46,910
so this is basically dividing by so 92 has
to be divided by 29 so the portion will be
275
00:36:46,910 --> 00:36:55,530
3 define so having obtain this we have a algorithm
so when we look at this so the last thing
276
00:36:55,530 --> 00:36:58,369
you have to
277
00:36:58,369 --> 00:37:05,859
divide by e so when you do that so this basically
saying 92 has to be divide by 29 so it has
278
00:37:05,859 --> 00:37:13,660
3 as portion and 5 a reminder so whatever
is the remainder is basically x ok so now
279
00:37:13,660 --> 00:37:25,560
let us see this equation so this is x so n
the given number n is 49 times x so +7 and
280
00:37:25,560 --> 00:37:35,440
the shows y also has 8 so finished so x is
5 and y is 8 so one thing which we need to
281
00:37:35,440 --> 00:37:40,720
understand with reference to this algorithm
is so once you have one solution you have
282
00:37:40,720 --> 00:37:42,040
infinite number of solutions (refer
283
00:37:42,040 --> 00:37:55,120
time: 37:46) so this is pretty evident se
for instance if x-alpha and y=beta is a solution
284
00:37:55,120 --> 00:38:22,540
is a solution 2 the equation that we had was
ahh by-c=ax then so x=alpha+bm and y=beta+am
285
00:38:22,540 --> 00:38:43,530
is also a solution m is any integer so this
can you easily check so where m is integer
286
00:38:43,530 --> 00:38:50,980
m so any choice of m has to satisfy and therefore
it is pretty evident that we have infinite
287
00:38:50,980 --> 00:39:00,570
number of solutions to this fine this is all
the algorithm all about kuttaka algorithm
288
00:39:00,570 --> 00:39:02,740
so now i have to need
289
00:39:02,740 --> 00:39:11,000
kuttaka algorithm and then move on in fact
what we have done is ahh later ahh of course
290
00:39:11,000 --> 00:39:15,450
brahmagupta has discuss this aryabhatta is
the first to discuss this kuttaka algorithm
291
00:39:15,450 --> 00:39:21,210
so brahmagupta has also discussed this so
bhaskara has done mahavira has done so all
292
00:39:21,210 --> 00:39:30,750
was later have provided certain kind of modified
version of the kuttaka and those will be covered
293
00:39:30,750 --> 00:39:40,660
when we discuss those text now i move on to
give you a flavour of how this
294
00:39:40,660 --> 00:39:49,160
jaina mathematics has been before of course
this mahaviracharya jaina tradition in jaina
295
00:39:49,160 --> 00:39:56,619
tradition ahh it seems that this mathematics
was also thought of considered as a part of
296
00:39:56,619 --> 00:40:10,359
their religious literature so we have this
jothisum as a part of vedhjangam so in fact
297
00:40:10,359 --> 00:40:16,000
it is called vedhanga jyotisha so basically
it gives some kind of mathematics so there
298
00:40:16,000 --> 00:40:23,910
it i think it is in a much more in fact they
say there is a section called in the religious
299
00:40:23,910 --> 00:40:26,020
literature itself so
300
00:40:26,020 --> 00:40:47,920
some of the ahh ancient jaina mathematical
works are listed here is a nice work so all
301
00:40:47,920 --> 00:40:55,660
of early jaina works are primarily based upon
commentaries are some of the original work
302
00:40:55,660 --> 00:41:04,260
and not even come to light so these 2 are
many of these was very unfortunate that even
303
00:41:04,260 --> 00:41:11,910
today that all the sophistication that we
have so not much k has been taken to preserve
304
00:41:11,910 --> 00:41:23,230
this manuscripts ok so results of mensuration
i just list a few of them if i have
305
00:41:23,230 --> 00:41:36,190
information the we do not have good manners
clips aryabhattiya fortunately we have plenty
306
00:41:36,190 --> 00:41:45,599
of and its commentary is only one and even
that is not acceptable so that is how things
307
00:41:45,599 --> 00:41:52,240
are so it is a very sad state of affairs anyway
so let us come to this jaina literature and
308
00:41:52,240 --> 00:41:59,150
some of this formula related to mensuration
so i am just listening here so for instance
309
00:41:59,150 --> 00:42:06,060
they say the circumference of circle is root
9 root 10 x metre so in fact brahmagupta out
310
00:42:06,060 --> 00:42:07,060
of
311
00:42:07,060 --> 00:42:13,300
uses this value so this is somewhat as you
know compare to the value which has been specified
312
00:42:13,300 --> 00:42:21,510
by aryabhatta anyway but for practical purposes
so this is how they have been using this so
313
00:42:21,510 --> 00:42:30,290
then area as a circle so you one fourth circumferencexdiamter
so this is all right but the value of circumference
314
00:42:30,290 --> 00:42:39,660
is not by that great so then there is a chord
and today morning we had a ahh discussion
315
00:42:39,660 --> 00:42:47,569
on great length of the aryabhatta sutra right
so that is all this is so
316
00:42:47,569 --> 00:43:01,370
basically the product of chord they are say
so so this is how it will be if you consider
317
00:43:01,370 --> 00:43:13,390
this as chord this is one this is another
so as far as this is bow and you consider
318
00:43:13,390 --> 00:43:19,210
this as bow this is so the product of this
will be the same as a product of these two
319
00:43:19,210 --> 00:43:24,450
so that is what it amounts to so this stated
in different way but that is what it amounts
320
00:43:24,450 --> 00:43:33,170
to so that is why 4 comes here so then sara
see once you know the diameter on chord
321
00:43:33,170 --> 00:43:39,119
you will be able to make the sara so which
this a cycle in certain way and there is various
322
00:43:39,119 --> 00:43:47,339
approximations which people have been trying
see to obtain sara given this sara and so
323
00:43:47,339 --> 00:43:57,210
on there are various formulas has been given
by jaina mathematicians so by sara so if you
324
00:43:57,210 --> 00:44:12,590
should understand that basically r-rcos theta
ok so give you a flavour of how the original
325
00:44:12,590 --> 00:44:24,290
verses are so i have just coated 2 verses
so this is may be in but definitely not in
326
00:44:24,290 --> 00:44:30,369
sanskrit and i have tried to make a sanskrit
rendering of this so that we try to understand
327
00:44:30,369 --> 00:44:57,770
so see here so this is ok 3 times the diameter
is circumference ok so c is 3xd then
328
00:44:57,770 --> 00:45:23,310
so this is 2 expressions one is the other
is 10 multiply by
329
00:45:23,310 --> 00:45:37,550
is diameter ok so d square so root of 10 times
d square is also that is what they have stated
330
00:45:37,550 --> 00:45:47,480
10 times the diameter so this ahh the expression
for the circumference once you know the diameter
331
00:45:47,480 --> 00:45:48,480
so
332
00:45:48,480 --> 00:46:04,640
these are the 2 different then in
philosophical literature we will see there
333
00:46:04,640 --> 00:46:17,650
are 4 states so one is and they call is the
fourth state so means the sense here uses
334
00:46:17,650 --> 00:46:33,720
one fourth understand so
is multiplication is diameter one fourth of
335
00:46:33,720 --> 00:46:55,450
that so that gives the ok what is a bit difficult
to more accurate ok so so this is the form
336
00:46:55,450 --> 00:47:03,910
in which we find some of these verses have
been cited by bhaskara also in his commentary
337
00:47:03,910 --> 00:47:06,690
2 aryabhattiya
338
00:47:06,690 --> 00:47:18,430
and most of the earlier in this language which
not for different from sanskrit but it requires
339
00:47:18,430 --> 00:47:36,910
a certain familiarity to figure out it
340
00:47:36,910 --> 00:47:56,089
and this literature similarly one more was
so so if you look at this so if you know the
341
00:47:56,089 --> 00:48:06,829
formula it easy to square root otherwise it
is not so easy so this so multiply by 4 and
342
00:48:06,829 --> 00:48:23,260
suppose this issue ahh then this should be
so chord square we can have this approximation
343
00:48:23,260 --> 00:48:36,059
and that is what
344
00:48:36,059 --> 00:48:37,059
he
345
00:48:37,059 --> 00:48:54,569
says then arc square this so this all sudden
approximations so which have been presented
346
00:48:54,569 --> 00:49:03,319
in jaina works so finally so this jaina also
has this notion of infinity express in 20
347
00:49:03,319 --> 00:49:09,940
different ways in fact they speak of different
kinds of infinity the countess motion of infinity
348
00:49:09,940 --> 00:49:15,119
in our mind and then what they have spoken
i mean that is not a good way then what is
349
00:49:15,119 --> 00:49:21,180
interesting is so they have spoken of different
kinds of infinity ok so we today
350
00:49:21,180 --> 00:49:27,190
we have to infinite number of infinity in
case so that is a different way so we have
351
00:49:27,190 --> 00:49:35,559
gone much more advanced way of analysing infinity
ahh here so they actually classify the number
352
00:49:35,559 --> 00:49:50,270
into is numerable is unnumerable and then
infinite and infinite also they say
353
00:49:50,270 --> 00:50:01,020
and then infinite in terms of area one direction
2 direction and so on and infinite everywhere
354
00:50:01,020 --> 00:50:10,109
so this is true perhaps is the something is
eternality fine so eternality in terms of
355
00:50:10,109 --> 00:50:11,109
space in
356
00:50:11,109 --> 00:50:19,660
terms of time in terms of direction fine this
are classify using various ways and
357
00:50:19,660 --> 00:50:28,069
some interesting examples which jaina have
chosen to convey what infinity is so for instance
358
00:50:28,069 --> 00:50:34,700
they say consider a trough whose diameter
is of the size of the earth ok so considerate
359
00:50:34,700 --> 00:50:45,140
a trough so which is 100000 so fill it up
with white mustard seeds so feeling a tough
360
00:50:45,140 --> 00:50:50,660
of the size of the earth and then the mustard
seed you have to fill it up and keep
361
00:50:50,660 --> 00:50:56,200
counting them similarly fill up with mustard
seeds other trough of the size of various
362
00:50:56,200 --> 00:51:02,930
lands and seeds ok so one is size of the earth
still it is difficult to reach the highest
363
00:51:02,930 --> 00:51:09,640
numeral so this just give you a certain conception
of what we are talking about and we see large
364
00:51:09,640 --> 00:51:24,710
numbers
ok so then we have also this very interesting
365
00:51:24,710 --> 00:51:31,099
thing so discuss about logarithm also but
it is very difficult to figure out see first
366
00:51:31,099 --> 00:51:33,210
of all i must admitted i am that
367
00:51:33,210 --> 00:51:38,809
conversion and secondly anyone could additions
and interpretations very very clearly written
368
00:51:38,809 --> 00:51:45,579
things are not available certain articles
are there but it is also based on certain
369
00:51:45,579 --> 00:51:55,530
other countries so point i want to convey
is so they so all that they discuss about
370
00:51:55,530 --> 00:52:00,670
various kinds of indices so it is in this
connection so people say that they also talked
371
00:52:00,670 --> 00:52:11,320
about this logarithms and i just skip this
and to give you an idea so what are the various
372
00:52:11,320 --> 00:52:12,320
topics which all
373
00:52:12,320 --> 00:52:19,740
discussed so in any difficulty in a text so
this might you have seen ahh professor sriram
374
00:52:19,740 --> 00:52:37,360
had discussed about mahaviracharya so combination
so were gone so on so with this i conclude
375
00:52:37,360 --> 00:52:42,300
my session on aryabhatta as well as jaina
mathematics ancient jaina mathematics ok so
376
00:52:42,300 --> 00:52:45,860
then you will have discussion on brahmagupta
and so on thank you