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so we had 2 lectures in aryabhattiya now i
will be leaving the third part of our discussion
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on aryabhatiya so if you recall ahh in the
second part so our lecture more or less ended
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with the discussion on the geometrical approach
to finding the sin table so i will start with
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that to recapitulate and then will proceed
with the analytic approach and this has been
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presented by aryabhata as impact 2 parallel
till even up to 15
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century because what he has finally done he
is the discrete version of what we know of
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harmonic equation today so that is what it
seems to be so it is very interesting thing
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so we will have a more detailed analysis on
this ahh lecture exclusively devoted for discussion
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on how the sin values have been improved upon
over a period of time then i will move on
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to certain interesting problems which have
been discussed by aryabhata so in connection
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with
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certain formulae which he presents with this
simple tool which we have been referring to
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us so that is what referred to us gnomonic
shadow so in samaskritham we call so based
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on that what are the things that can be found
we will discuss an application in astronomy
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and then even day to day basis suppose if
a lamp so if you want to find out the height
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of the lamp if you want to find out the distance
of the lamp so how does this device tell so
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in finding all
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these quantities which will be of practical
use so we will do that then we will move on
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to the discussion of the famous of course
it has been discussed in greater detail but
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what i am going to do is basically choose
some very interesting problems which have
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been given by bhaskara in connection with
that so here also notice what i have written
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is jya-sara-samvarga-nyaya so jua as i said
is the card semi card so refers to the burner
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so is product so here
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it is basically the product of the card he
says since we know in a circle so what are
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the applications of this rule so that is what
i will be dealing with that in great detail
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and in this connection bhaskara has presented
very interesting problem as illustration so
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those will also be dealt with so the hawk-rat
problem hawk-rat problem with bamboo problem
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the lotus problem the fish-cane problem so
all that will be highlighted today then i
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will move on to the
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arithmetic progression and sum of these so
to recapture what we did so this geometric
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approach essentially had a simple observation
that to the card length of one sixth of the
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circumference is going to be radius so with
this the entire table was constructed once
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you know that the i told you that the radius
so is determined based on the value of pi
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which is given by so once that is known 3438
so then we know r sin30 so rsin 30 is known
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therefore
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this cd is also known or card circle is known
and then r-r cos 30 is also known so bc is
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known cd is known and therefore this hypotenuse
bd is known and from bd you get sin 15 degree
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so sin 30 is known sin 15 is known and the
general principle sin theta to cos theta cos
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theta to verse theta and from this to you
get sin theta/2 so this principal and so as
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i showed you ahh so the table goes like this
so once you know r sin 90 and then with sin
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30 we
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will be able to get almost 15 values ok so
this is what these 3 is and this 3 present
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8 values of 15+323 and sin90 this si will
all the 24 values of sin so what is interesting
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to note here is so in geometric progression
mean you measure so it is in some sense so
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you observe the geometry and then you get
it so that is what it is all about and you
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do not do any measurement taking a rope and
then determine the sin value that does not
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mean that all that is
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required in this basically ahh the technique
for obtaining square and square root which
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has been thoroughly discussed by aryabhatta
and so you will be able to generate the sin
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table so this is what it is now i move onto
then another verse which presents the analytic
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expression with which it is a sort of recursion
relation so based on which will be able to
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get the entire sin table constructor the verse
goes like this
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in that this is one of the most
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thus verse we can be found in aryabhatiya
and 16 difficult for figuring out of it but
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for the help of the commentator so in fact
this verse has been slightly differently interpreted
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by different commentator so bhaskara has given
a certain ahh understanding to us and neelkanth
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slightly different presence this for ultimately
all of them how is this relation to be extracted
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out of this verse this has been slightly different
approaches so now i am not going
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to discuss that in great detail all that i
want to say is this verse basically translate
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into this equation so if you look at suppose
i know value of sin theta so theta here is
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the certain unit and this unit as i mention
so if you divide the quadron 24 this happens
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to be 225 minutes so all that you do is sin
theta is theta you just take as a approximation
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so you take sin 225 minutes is same as 225
ok so i will just take this then how do you
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generate the entire table so
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the relation with has been presented by aryabhatta
amounts to this so rsin theta is known here
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and for any value you will be able to generate
once the first value is known using this relation
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if you take i=1 and you want to get i+1 right
this is how the recursion relation is used
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so you know for first value now you have to
study suppose it is i is 1 so then what is
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i+1 theta sin 2 theta so we will be so in
this relation so this is 0 right so and we
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have sin theta here so the sin
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theta-sin theta/sin theta so -1 so the second
value will be 224 the first value is 225 second
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value will be 224 so once more the second
value you use this recursion relation you
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will get the entire sin table so this is the
method which has been method which has been
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presented so the ahh method which has been
presented by aryabhatta so is to essentially
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get the find difference table so what he does
is from the previous value see so this is
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previous value these
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2 terms put together so you have this previous
value of sin theta so divide this this factory
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same so this is 1/225 is what i am but i have
chosen the exact recursion relation this is
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easily obtained one can show that the factor
which we will have here is 2*1-cos theta so
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later astronomer so this i will discuss in
greater detail in a separate lecture so they
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have chosen the value see 1/225 amounts to
this and the other astronomers have given
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this value is exact value
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is this an as improvement so this has been
ahh defined instead of 1/225 so the astronomers
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in the kerala school have you chosen the constant
to be 233 and half ok that will be very very
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close to this almost 6 to 7 decimal place
accurate so how did they do extra will become
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clear when we move on to the kerala school
our discussion on kerala school will make
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everything clear so with this i just want
to make one observation as regards this method
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which has been presented by aryabhatta this
analytic approach to get the sin table so
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there is an interesting observation which
has been made by dalamnre he says the method
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which is curious he refers to the ahh difference
sin difference table which has been obtain
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just now discuss now about this he says this
method is curious it indicates a method of
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calculating the table of sin by means of their
second differences so if you note this so
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this is basically first
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difference and first difference so you find
the difference of these two first difference
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all that you note is it is proportional to
sin so this is what i meant by saying that
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this amounts to the difference equation second
order differential equation ok that is what
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he has in his mind when he says that by means
of second differences the differential process
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has not approve been employed except by bricks
so he is talking about in 16 or 17 century
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and who himself did not
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know when the constant factor was the square
of the chord here then is the method with
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indians possessed and which is found neither
amongst the greeks nor amongst the arabs so
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this is something which i need now i move
on to problem s please quit mini now i move
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on to problems related to this shanku so this
si the verse he present the certain formula
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so which has to do with the shadow how to
obtain the shadow length and so
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on so in this figure if you note xy is ok
this is our device with which we carry on
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the experiment and here ac represents the
lamp post so here this the term has to be
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understood just keep in mind because we will
have an occasion to ahh recall this and therefore
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i have given the note which has been given
by bhaskara here he says is a lamp lamp post
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height of it so this of course it can be 1
seconds in a right angle triangle is
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this can be chosen as good as this can be
chosen as does not matter but here so they
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have created a certain connection and disconnection
has been made clear by bhaskara in his commentary
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because this should is this and is this and
basically means distance or separation what
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is stated in verse is is xy is multiplication
take a product product of distance of separation
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ax is what is used to find the ahh difference
between 2
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quantities so normally means speciality but
in the context of mathematical text many times
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you will find the to be used to refer to the
difference so basically is the difference
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in the height of lam post so this is what
it is ac is so the height of that and fi is
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the difference between 2 is division fine
so whatever obtain is so he says this is basically
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the shadow of the so this verse has been stated
by aryabhatta in order to
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ahh obtain the shadow that is cast by ok the
length of the shadow so which means if you
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know these two things xy and ac so then you
will be able to get this but this has a certain
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astronomical application and that is why he
has given this relation so otherwise i am
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in this is just based on so two similar triangles
and you will be able to get this relation
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so formulas from the base of the where is
going to be the tip of the shadow so
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this is what this gives so this is a simple
straight forward application of considering
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two similar triangles so the application comes
here in the case of lunar eclipse so you can
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think of you just see this so this is basically
a depiction of what is happening in a lunar
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eclipse so the moon enters into the shadow
and earth is in between the sun and the moon
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and if you just think of cutting this into
half so basically what you have in this kind
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of a setup so here the
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can be taken to be the radius of the earth
semi diameter of the earth and this this object
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ac is semi diameter of the sun so basically
the ray from this stop so as it moves and
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in goes then which as this point so if you
know this is height and this height then you
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will be able to get the shadow this is what
basically aryabhatta said and this is an application
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and wherein you know the semi diameter of
earth you know the semi diameter of sun so
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all this will be specified and with this you
will be able to and how do you know the distance
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between sun
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and earth so that is see in terms of set another
it is not the exact distance but we will be
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able to get from the what is known as this
ok this is one application the other application
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is so privacy practical suppose you have lam
post and if you want to find out the height
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of the lamp post so you can actually measure
the shadow what is there was problem so you
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know the shadow you measure the shadow and
hence you can calculate any other quantity
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so this is
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another this this this this problem has been
ahh given as an illustration by bhaskara which
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is so found the length of the shadow to be
16 is the lamp post is 72 72 is the height
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of this post say so what is this distance
ray at so you have to specify this is just
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illustration so you know one you get the other
this is ok so we will move on to other interesting
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verse in aryabhatiya and in fact the commentary
to this verse by bhaskara is
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something which is very very ahh very fine
i would say i wiull discuss that so the problem
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here is to find out i mean one one one important
application is to find out the height of a
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lamp post this is consistent considered to
be a general problem of this nature and here
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what he say you carry on the experiment twice
so once keep the in one place move the to
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another place so and then measure the shadows
with cash so from that so what is it that
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so the formulation of the problem is this
way let me explain this with
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equation so is the length of the shadow ok
so so the length of the shadow so he refers
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to mc as well as qd so both are since you
are carrying our twice then is tip of the
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shadow is separation so so if you consider
these two shadows basically it represents
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cd so c is 1 and d is another so basically
is cd is
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basically subtraction but what so aryabhatta
has not stated but this has to be understood
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and here so this happens to be
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the difference in the ok so you have to subtract
so is the remainder ok so that is what is
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mentioned by aryabhatta by the word divided
so what it gives actually dual usage and what
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does this mean see that is why i said you
were remember the word was used to refer to
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the height of the lamp post so that is then
obviously the perpendicular is and the perpendicular
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here refers to the plane here ok so this is
what is and here there will be 2 and this
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is if you conduct the experiment by placing
the at m then
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bc is 1 if you conduct the experiment by facing
some quote q then bd is both of them are depending
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on where you place the and that is why we
have the dual usage so so this formulation
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which aryabhatta has given for the expression
of so you get 2 expression for one is for
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bd ahh the other is bc so look at this now
and connect it with the verse so is qd this
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is one and this is another so then when he
says you multiply
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this so pq and lm in fact both are ok we are
at the same dimension obviously when you conduct
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this experiment divide by corresponding so
if you then that will give so did i use this
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expression and use this expression and we
will be able to get the height of the lamp
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post bhaskara present some interesting discussion
on the property of the application of this
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rule to find the distance of separation between
sun and the earth so one can think of sun
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be a sort of lamp like lamp post showed is
there so we can use this so
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you can do can we conduct an experiment and
then get the distance of the sun from the
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earth so this is what you so means a certain
day it is actually referred to as equinoctial
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day the sun actually moves from ahh equinox
towards the north and then again towards the
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south so this what we call as but suppose
you conduct the experiment on equinoctial
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day so so he says so imagine that the sun
is exactly on the prime meridian so sun on
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prime meridian
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so the shadow will be exactly
in the north south line ok so
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is so just now we had the discussion of finding
the difference in the right so difference
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between
the so you can find so you have so all this
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so they try to do this so so they thought
that they can find out the distance of separation
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between the earth and the sun so by conducting
this kind of an experiment ok so then he says
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means that is improper in fact he says in
a
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given to conceive of to speak of so that mean
i can think of 2 spacing and 2 different spaces
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and then measured so this is not actually
going to work out
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it just goes on but the point is what we need
to understand here is this is something which
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is of astronomical magnitude distance and
the entire this like a point object so we
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just think of placing a place it here or you
place it in a in mumbai or replace it elsewhere
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so it is not going to make
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any sort of different so this is not going
to work out so that is what i think bhaskara
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wants to imply that you cannot just think
of using this experiment to find out the distance
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between the earth and the sun this is something
which is obviously possible for distances
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which is of terrestrial magnitude and not
celestial magnitude so this is what is convey
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then we have 2 so which is presented in one
arya is a certain theorem we can take it
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via general principle is called so he says
so this is something which is very well known
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so so think of this so just make the terms
familiar once more i just think of this triangle
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oae ae is oe is and ao is karna so he says
oe is square b square oa square ok so this
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is what he says then in the later half of
the verse which i have referred to as ae and
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00:27:13,220 --> 00:27:22,650
ec so the product of ae and ec means ec=ae
therefore it is ae square so which is same
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00:27:22,650 --> 00:27:23,650
as
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00:27:23,650 --> 00:27:34,140
the product of de and eb is basically arrow
so if you conceive this part of the circle
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abc then eb is consider the other part of
the circle adc then de so all that he says
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is the product of product of chords is equal
ok this has been illustrated with very interesting
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problems by bhaskara so i thought i will just
spend a few minutes on that this application
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of this has been illustrated in various contacts
so which look quite apparently different but
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then
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the principal which has to be employed is
one and the same so that has been very beautifully
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brought out by bhaskara by giving very interesting
examples the first problem is this this hawk-rat
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problem so bhaskara says means in this connection
while trying to explain this principle and
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its applications discuss in great detail so
first as an introduction he just says if you
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look at this is is centre o is this so this
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is
so
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keeping this problem which is going to be
stated so he says is this ok is the height
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of that so the height of the pool so now the
problem is this ok the problem is the following
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so let us take this to be the pole so this
is sitting here and
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so this is given to be 18 so here is a hole
at the bottom of this pole and then he says
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some rat
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which came out to the hole so it is somewhere
here so this distance is he state it as i
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think 81 so the circle i just want to draw
only for the application of this ahh direct
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00:31:46,990 --> 00:31:56,440
application of this principle so he says so
this rat observed this half and then it got
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scared and it wanted to just run into the
hole ok this what he says so which has come
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out out of here so it want to get into get
into is what is called so it just starts moving
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00:32:17,900 --> 00:32:21,010
here so this rat
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but the moment we saw it also flew down and
then grab the rat here and it finished it
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00:32:32,240 --> 00:32:52,360
so now the question is the distance travelled
by means to how much it has to cover in order
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to reach the hole so so what was the distance
so basically the question is what is this
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and what is this assuming that both of them
travel at the same speed so this is the problem
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00:33:13,350 --> 00:33:23,090
and this one can let me see that it is your
application of this which has been discussed
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00:33:23,090 --> 00:33:24,760
so this distance
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and this distance on the same because it travel
at the same speed so we have this suppose
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you see same location that we use here so
ea so let us take ea ahh so ea square/call
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00:33:51,210 --> 00:34:22,980
the later d so d is eb so ea is known de so
that is all you will be able to solve this
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00:34:22,980 --> 00:34:37,499
problem ok this si a direct application of
this and very different problems more or less
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00:34:37,499 --> 00:34:48,499
they use the same principle
in fact the answer are 38 and half and so
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00:34:48,499 --> 00:34:55,259
this is basically so this you know so this
able to
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00:34:55,259 --> 00:35:07,950
solve this problem and what you will basically
get a this 81+4 so this will be 4 to do that
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00:35:07,950 --> 00:35:16,609
so 81+4/2 so and this will be the radius and
81-4/2 so that will be this is what to ask
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00:35:16,609 --> 00:35:40,740
ok so this is the answer then we have this
bamboo problem so
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00:35:40,740 --> 00:35:57,269
think of a bamboo
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00:35:57,269 --> 00:36:15,650
and bamboo so the height is 16 from the base
the distance is 8 so tell me so a given height
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00:36:15,650 --> 00:36:31,519
and he tells you so the distance of the tip
from the base so tell me this is a problem
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00:36:31,519 --> 00:36:33,200
so this also
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00:36:33,200 --> 00:36:50,769
very very similar to this identical in fact
so one should think of this to be the
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00:36:50,769 --> 00:37:03,680
so this is our bamboo and this distance so
it stated to be 8 i think so the tip the tip
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of the bamboo so this bamboo is like this
so this broke and top like this so this is
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00:37:15,200 --> 00:37:24,140
specified and this gives this you should be
able to get the desired answers the answers
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00:37:24,140 --> 00:37:34,299
are 10 and 6 ok so 10 to the top and 6
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00:37:34,299 --> 00:37:44,490
from so the total high total high was stated
to be 16 and the third problem is lotus so
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00:37:44,490 --> 00:38:03,380
there is a lot of discussion which goes on
as to how to make mathematics learning clear
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00:38:03,380 --> 00:38:09,910
so that are very interesting problem which
one can find in this so illustrate very place
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00:38:09,910 --> 00:38:16,839
principle and application of principal in
a wide variety of problems so this is again
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00:38:16,839 --> 00:38:23,829
a completely different kind of a problem that
you will see that it also today so the problem
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00:38:23,829 --> 00:38:25,660
that he stated here is
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00:38:25,660 --> 00:38:55,150
the following so he says that somebody noted
the kamala kind of a lotus and this is all
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00:38:55,150 --> 00:39:10,069
water if he says so this is based upon so
i do once again the circle only to show that
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00:39:10,069 --> 00:39:20,460
we make use of the same principle so all that
he says is there was a wind which was flowing
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00:39:20,460 --> 00:39:44,990
mildly so in this direction and this sort
of so this flower got from here
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00:39:44,990 --> 00:39:54,549
and it is like so this all he says tell me
what is the height of the kamala and what
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00:39:54,549 --> 00:39:55,549
is the height of
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00:39:55,549 --> 00:40:09,380
the water so what is the data that has been
given so the data given is this is 8 and this
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00:40:09,380 --> 00:40:33,150
distance so he specify it as ahh is
basically 24 ok so is 24 so this once again
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00:40:33,150 --> 00:40:44,420
you can see that it is the same principal
we can see that so if imagine this problem
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00:40:44,420 --> 00:40:55,599
is essentially so this is stated so which
is like gives this sometime he gives this
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00:40:55,599 --> 00:41:02,009
distance so in the case of hwk-rat problem
so this distance was given now he is giving
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00:41:02,009 --> 00:41:04,720
this distance and in another problem he
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00:41:04,720 --> 00:41:10,779
gave this distance so it is all problems with
very different the principal is one and the
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00:41:10,779 --> 00:41:27,119
same so the moment use this now i will be
able to get solution to
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00:41:27,119 --> 00:41:35,510
this problems a no no when it was when it
was vertical so it is 8 above the water yeah
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00:41:35,510 --> 00:41:53,230
no no it is so here refer to from the original
so
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00:41:53,230 --> 00:42:03,320
this distance is 24 what is height of kamala
and what is the height of water so the answer
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00:42:03,320 --> 00:42:07,619
you can easily see the 40 and 32 everything
is used
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00:42:07,619 --> 00:42:12,000
on the same principle ok finally quickly so
let us even more problem and then i will proceed
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00:42:12,000 --> 00:42:28,380
further ahh so this is small error here this
is not another kind of problem fish and crane
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00:42:28,380 --> 00:42:36,099
so this problem is stated like this so is
a slight variation of this problem not exactly
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00:42:36,099 --> 00:42:40,260
identical but he presents this way i use the
same figure so we should imagine a certain
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00:42:40,260 --> 00:42:46,349
pond so this is what is referred to as ( so
the dimension of this so
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00:42:46,349 --> 00:42:50,799
this is the dimension of is stated to be 6
and 12 suppose normally in all this kind of
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00:42:50,799 --> 00:42:53,009
descriptions where you have to have geometrical
figures drawn so the always specification
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00:42:53,009 --> 00:42:57,410
direction ok so we just take it to be this
suppose to east and this is north ok so he
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00:42:57,410 --> 00:44:08,769
says is this fishes so the fish is here in
this corner of
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00:44:08,769 --> 00:44:35,829
this pond so is northeast
actually is north west ok so is standing here
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00:44:35,829 --> 00:44:54,029
ok
this is the bird and he thought that he should
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00:44:54,029 --> 00:44:55,029
slightly
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00:44:55,029 --> 00:45:11,630
escape from there so he says cutting across
so this is north so what he says is so this
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00:45:11,630 --> 00:45:23,060
fish try to come to the southern side but
that time also walk so it noted that he is
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00:45:23,060 --> 00:45:37,650
moving and this all went like this so along
the so now he is asking so so what is the
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00:45:37,650 --> 00:45:43,660
distance travelled by these things so once
again you will see that this application of
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00:45:43,660 --> 00:45:53,289
the same thing so without so he only thing
is what is be understood here is this distance
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00:45:53,289 --> 00:45:54,289
and this
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00:45:54,289 --> 00:46:05,829
distance are one on the same so shifted it
and you will get the same problem and this
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00:46:05,829 --> 00:46:11,450
of the different kinds of interesting problems
which are presented by bhaskara as illustration
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00:46:11,450 --> 00:46:25,400
of these fundamental theorem ok then i will
very quickly discuss to show how aryabhatta
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00:46:25,400 --> 00:46:37,190
has been able to present the formula in very
interesting form of composition so this arithmetic
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00:46:37,190 --> 00:46:43,579
progression problem so this is all well known
result but it is only the the
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00:46:43,579 --> 00:46:53,069
form of the languages with wanted to convey
and how bhaskara has interpreter this bhaskara
302
00:46:53,069 --> 00:47:00,599
says present 2 3 expressions so it is not
necessarily a single expression but you have
303
00:47:00,599 --> 00:47:07,230
to appropriately combined the words which
has been presented in this to get different
304
00:47:07,230 --> 00:47:13,739
formulae so this is what it is only to show
what kind of thing i am just putting this
305
00:47:13,739 --> 00:47:21,569
version the commentary so bhaskara in fact
says before commenting up on the verse is
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00:47:21,569 --> 00:47:26,329
a certain
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00:47:26,329 --> 00:47:38,190
formula many formula have been given by aryabhatta
in this single verse this is a certain style
308
00:47:38,190 --> 00:47:45,299
of composition see in sanskrit so supposed
2 3 versus which are combined then it is called
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00:47:45,299 --> 00:47:52,660
so this 2 verse have to be read together in
order to get the meaning so in kalidasa and
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00:47:52,660 --> 00:47:58,380
then some 4 5 verses will be there so all
the surface that we put together in order
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00:47:58,380 --> 00:48:04,019
to understand the only one verb so which runs
through all this process but is opposite kind
312
00:48:04,019 --> 00:48:09,470
of a thing so within a single thing so it
can combine independently means to be kind
313
00:48:09,470 --> 00:48:15,099
of a thing so freely can order this word to
get 1 formula you can freely order some other
314
00:48:15,099 --> 00:48:20,539
words and then you will get another formula
so that is why he is saying means they have
315
00:48:20,539 --> 00:48:31,460
been arranged in the format have to put them
together to get the formulae is just to give
316
00:48:31,460 --> 00:48:49,729
you a flavour i thought i should do this so
this is with reference to
317
00:48:49,729 --> 00:48:57,950
an arithmetic series so think of an arithmetic
series a+ a+2d and + so on so this formula
318
00:48:57,950 --> 00:49:08,509
he says is means one is removed from that
so basically is a number of terms in this
319
00:49:08,509 --> 00:49:15,910
is so remove one divided by two so common
difference so by factor which with increase
320
00:49:15,910 --> 00:49:28,410
so is the first term for the first time so
add the first term so what does we use this
321
00:49:28,410 --> 00:49:44,190
give you the so he just have to combine this
value of the seires fine so then he
322
00:49:44,190 --> 00:49:58,140
says so this so whatever you are saying so
that you take to be the other part so it gives
323
00:49:58,140 --> 00:50:10,200
you the is sum of n terms ok so this multiplied
by n gives you the different things so is
324
00:50:10,200 --> 00:50:20,059
the first term and is the last term so which
is divide by 2 so then also you get the so
325
00:50:20,059 --> 00:50:24,700
this are in fact much more interesting thing
has been discussed by bhaskara but just i
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00:50:24,700 --> 00:50:26,239
will stop here thank you