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so this is the second part of our lecture
on aryabhatiya so in the earlier part so we
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saw so the algorithm presented by aryabhatta
for extracting square root cube root and so
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00:00:26,130 --> 00:00:35,320
on so in this lecture we will be starting
with the formula given by aryabhatta of finding
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00:00:35,320 --> 00:00:43,690
the area of various geometrical objects so
starting with triangle then proceeding towards
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trapezium circle and so on so in the latter
part so we will see the
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00:00:50,450 --> 00:00:58,290
approximation which has been given by aryabhatta
for the value of pie and the method which
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has been by him for generating the sin table
so in fact aryabhatta has given 2 different
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00:01:09,500 --> 00:01:15,180
approaches for constructing sin table so one
is the geometrical approach the other is the
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00:01:15,180 --> 00:01:23,650
analytic approach so we will see both of them
and then ahh during our discussion on the
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verse which gives the value of pi so we also
see an interesting note which has been given
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by
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the commentator bhaskara on the use of the
word asana so this has been discussed at great
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length given by nilakantha somayaji which
will be covered later but here we will see
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the note which has been given by bhaskara
on used the word asana so let us start with
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the verse which presents the formula for the
area of a triangle so aryabhatta says so normally
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he say half into this into height so it is
essentially same expression which has been
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presented is the word which has been employed
by people to refer to the area so here the
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word should be understood as the measure of
the area so normally we are familiar with
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half into base into height so here this can
be taken to be this site is base and is the
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height ok so in fact in the word has been
analysed by the commentator bhaskara see in
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a triangle usually ahh if you have a right
angle triangle so let us
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say abc so if ab is then ac is and of course
this is called karna so if this can we interchange
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dividing up on the size of the angle so one
is and the third is karna now if we consider
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a triangle so this is the part which we are
calling as so and this is so this is the base
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this is height half of it base is the area
of the triangle in fact the word gala is generally
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00:04:27,750 --> 00:04:36,740
used to refer to half splitting ok so means
so so generally is taken to be
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00:04:36,740 --> 00:04:44,330
equal division ok so when you thought of divide
so this is how the meaning as such comes from
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that but then if we use this then if not be
applicable for all the triangles in fact bhaskara
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so comment is discussion by classifying the
triangles into three types he says they are
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all synonyms and there are 3 type of triangles
further continuing the discussion he says
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in
fact we should have 3 types of triangles
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this is applicable to all the triangle or
is it something which is specific to one of
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the three types or any of the 3 types so he
says this means so the group of all the class
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so it is applicable across the class of
the triangles so then he goes on to discuss
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about the word so if you say so if you use
this kind of derivation of the word then it
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00:06:12,440 --> 00:06:22,040
is confined to only these two types ok so
so if you draw perpendicular then the perpendicular
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device the
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00:06:23,040 --> 00:06:29,110
base into two equal half only in these two
cases scaling type but this formula which
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00:06:29,110 --> 00:06:33,970
has been presented for area is applicable
to all types then how do you understand the
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word so this is the question which has been
raised by bhaskar and he actually quotes very
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interesting matching i would say so the word
ahh should be considered as a so in the sense
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so if you not so try to find the derived meaning
of the word and it has to be just for
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00:07:03,630 --> 00:07:11,340
instance so in the case of the word in sanskrit
so is generally referred to as a goat but
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00:07:11,340 --> 00:07:17,370
if you go to the derived meaning that which
is not known so it never applicable to goat
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00:07:17,370 --> 00:07:23,639
ok so when you consider this so you just have
to accept whatever is the meaning which has
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00:07:23,639 --> 00:07:29,199
been accepted by the society at that point
that is what he is trying to convey so if
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at all you want to provide some kind of a
is not for the purposes of but this is the
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very
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00:07:37,830 --> 00:07:43,380
interesting matching so which can be applied
in various disciplines various contacts here
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the categorically states and therefore this
derivation should not be taken in the literal
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sense here so should be simply understood
as the perpendicular that is dropped on the
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base from the vertex so this is what the sutra
aryabhatta sutra means so the area of the
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triangle is so half of the base into height
so this is all it is so it is applicable to
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all the three types of triangles so this verse
basically tells you that the area can be calculated
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using this formula but then if you do not
know the height so you only know the sides
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of the triangle then how do you go about finding
the area so then there should be a way to
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find out the height of the triangle so that
is what is discussed by bhaskara in his commentary
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so the word has been employed to refer to
the two parts which are generated
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by dropping the perpendicular from the vertex
ok to drop the perpendicular which is referred
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to as so bhaskar essentially gives a certain
method by which you force to calculate the
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and then from that you find out the so if
you know only the sides so you know the base
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so that is taken for granted and for finding
height so based on the 3 size for the presence
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of formula this is how it goes so first he
gives an expression for the difference in
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the abacus and
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00:09:41,519 --> 00:09:55,639
the difference in so bd-cd bd is cd is so
the difference is c square-b square/Armstrong
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00:09:55,639 --> 00:10:03,239
so this can be easily seen and we have an
expression for the difference in in terms
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of the sides of this ok so if you have this
basically so abc
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so this is b so bd+cd=a which is known and
bd-cd if given to be c square-b square in
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this so once we have these 2 we have the expression
for so just add this and subtract this so
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you have the
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expression for the 2 so this is stated to
be half of is the word which is used to refer
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to the base so difference so this gives you
the expression for both the so having known
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this so it since you see you can calculate
p either from this or from the other so this
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is basically so the karna can be either c
or b depending upon either of the triangle
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that we choose so this is what ( ahh in fact
is a very interesting to read
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the way he present this c square-b square
is putting together essentially means sum
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and is difference so he says bsquare-c square+c=b
and there is also equal to so the sum of the
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two in the difference to the 2 so this is
straight away seen from the conservation of
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these two triangles ok so you get this so
this is the first relation at he says so then
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so a from this expression we see that so a
difference is so csquare-b square so which
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what he stated in the earliest slide and therefore
is this so this is how bhaskara proceeds in
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fact the sides verse so whenever you have
something of this form this is called and
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this is called so you have you have to just
sub this then find the difference in fact
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00:12:57,750 --> 00:13:11,809
this is how demonstrated with the formula
example so
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this is the bhaskara from several examples
on various occasions wherever he has to illustrate
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the use of a
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particular formula so i will skip this basically
ahh the presents how one finds the area of
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a triangle so given the three sides and you
have to calculate the height and from you
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have to get the areas so now i move on to
the area of circle so which has been presented
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00:13:44,870 --> 00:13:59,669
by aryabhatta this is the very interesting
so in fact it is half verse wherein ahh ahh
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he has used the word has lot of significant
refers to circumference so is diameter
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so is half of it so so it refers to half the
circumference so the arc has to be absolutely
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significant thus far it means so
means multiply by the radius semi diameter
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so this is the fine so this pi*r square fine
so commenting on the usage of the word so
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which is quite edifying so he says so why
did aryabhatta used the word so this is meaningless
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or can be assign from sense to that so this
is the analysis which it has taking
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a very simplistic one can say so in certain
cases so we have to fulfil the metrical compulsions
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and therefore so one add some word which need
not be necessarily conveying something which
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is very meaningful so this is one way of saying
that has been used in just to feel the metre
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of the bus so putting a
certain constraint on that means which you
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can involve so finding the area so is there
any other means no no there is no that you
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only this way so that is
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the way this has to be understood so there
is no other means so this is the only mean
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to obtain the exact area of a circle so what
should he say so if there are different path
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you reach the place then you can say choose
only this path there is only one path then
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there is no point in saying that you have
to go only with path so then he stars he ask
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this question no no this does not seem to
be appropriate that also means
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so what is so somewhere it stated
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so radius is square ahh so r square multiplied
by 3 so which means the value of pi is approximately
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getting to be 3 and for various practical
purposes people having employing this so in
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order to see that those things are not taken
to be the right formula to be employed for
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00:17:20,819 --> 00:17:30,860
finding the area of a triangle so this aryabhatta
has given a certain guide line that this is
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the to obtain the exact area so where you
do not require that kind
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of accuracy so next we move on to the area
of trapezium ahh ahh see in fact the latter
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half of the word actually presents the area
of a trapezium and the earlier half of the
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verse has been used to define something else
is a certain geometrical figure so that is
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the sense which has been used this is the
area of this khetra here is used to refer
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to the base and the face ok in this figure
bc is base and ad is the face is
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multiplying so what is the other term
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ok so is used to refer to this perpendicular
distance the distance of separation between
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the base and the face okay so the area of
trapezium is halftime so this is the formula
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which is used so what is the earlier part
of the verse so here he says distance i said
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so which is mark as p here is multiplied by
the so is also basically the site ok so so
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this bc and ad are also referred to as 2 part
so so this f*p and d*p so yoga is again refers
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to the some of the base and face that 2 part
was refers to c and b so it consider 2 diagonals
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drawn so then this is the formula that has
been presented in fact the next was a very
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00:20:40,450 --> 00:20:52,059
interesting verse wherein a analysis has been
done and these analysis has been done by bhaskara
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which is quite verifying when we look at his
commentary he says generalise approach to
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finding area so initially started triangle
anymore on trapezium when it
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discussion about circle so when he says either
a way general way of understanding how we
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compute area when we see some kind of a similarity
or can we do something so there can be general
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formula prescription can be given and you
can fit in different cases into this by and
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considered as a special case then only thing
that is kind of analysis which has been done
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by bhaskara while coming up on this verse
so geometrical figure geometrical figure
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so you have to just find the side of appropriate
value of side take a product of this so area
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is product of two things so with this in mind
this has to be different appropriately for
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various objects so that is the import of this
is what is conveyed by bhaskara so is area
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is the product after all having given expression
independently for all this so why is aryabhatta
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give one more word so this seem redundant
so do not need so this is the kind of question
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00:22:30,149 --> 00:22:31,149
that
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00:22:31,149 --> 00:22:37,440
he rises and then he says so this can be considered
as a different way of arriving at the same
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result sort of cross check one with the other
ok so that is the purpose of this and if the
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00:22:46,230 --> 00:23:00,230
commentary goes like this basically computing
so so this depending upon the case which you
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are ahh parentally handling depending on the
circle trapezium this whatever so you have
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00:23:07,000 --> 00:23:20,840
to appropriately find the size in fact he
goes on let us
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00:23:20,840 --> 00:23:34,860
consider this figure so in this ok is basically
can be defined as product of 3 things so what
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00:23:34,860 --> 00:23:48,669
is that half of this so in the case of ahh
the triangle so what happens to the face so
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00:23:48,669 --> 00:24:08,340
i just collapses to 0 right so 0 in the case
of rectangle so same as b ok and how do we
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go about for circle he says is the kind of
distance separation between the faces so in
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00:24:21,360 --> 00:24:37,260
the case of a circle what you have to conceive
is is to be taken as so
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00:24:37,260 --> 00:24:49,830
this is the kind of visually fit to be a rectangle
even this circle can be conceived of this
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was has been explained by bhaskara in fact
further there is a very interesting discussion
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00:25:00,629 --> 00:25:09,690
so i want to just code this a certain maximus
which has been employed by bhaskara to justify
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00:25:09,690 --> 00:25:23,029
the presence of this verse here ok so in tamil
they say if you throw one stone you get 2
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00:25:23,029 --> 00:25:28,509
mangoes oru kalula rendu manga so something
like this so in one
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00:25:28,509 --> 00:26:00,980
effort finding the area and prathikaram is
verification ok for a
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00:26:00,980 --> 00:26:07,999
particular purpose in mind but it can also
serve certain other purposes so this quotation
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00:26:07,999 --> 00:26:31,509
is quite interesting means water canal ok
so it is a is basically form of rice ok water
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00:26:31,509 --> 00:26:41,139
canals are created to water the fields so
incidentally somebody can go and use the water
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00:26:41,139 --> 00:26:47,029
for some other purpose so nothing is lost
so it is in this sense so we can understand
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00:26:47,029 --> 00:26:48,029
that
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00:26:48,029 --> 00:26:57,700
this sloga so which has been composed by aryabhatta
to cancel several purposes so now i move on
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00:26:57,700 --> 00:27:08,380
to get another interesting topic which actually
forms the basis for one of the construction
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00:27:08,380 --> 00:27:22,820
methods which has been described by aryabhatta
for finding the time table so this half verse
165
00:27:22,820 --> 00:27:38,249
this is the statement is circumference half
of the circumference refers to the god ok
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00:27:38,249 --> 00:27:58,970
so in fact suppose you consider circle ab
is a
167
00:27:58,970 --> 00:28:07,600
part so which is referred to as so this portion
of the circle so this is referred to as which
168
00:28:07,600 --> 00:28:27,259
actually means bow so this looks like arch
looks like bow and any arc length is referred
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00:28:27,259 --> 00:28:49,190
to as or any synonym for that so this card
a b is refer to as various terms which have
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00:28:49,190 --> 00:28:58,179
been used so this is out of conceived as bow
and this is considered as string and they
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00:28:58,179 --> 00:29:15,320
all mean string so what is stated in this
work is so one sixth ok so if you conceive
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00:29:15,320 --> 00:29:16,320
a
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00:29:16,320 --> 00:29:24,940
circle so divided into 6 equal parts so the
card corresponding to one sixth of the circumference
174
00:29:24,940 --> 00:29:35,389
is referred to as a and he just states so
this is same radius so this is pretty evident
175
00:29:35,389 --> 00:29:48,110
so if you think of this triangle obc so which
is an equilateral triangle so we have this
176
00:29:48,110 --> 00:29:58,559
cards which is same sa the radius fine so
card of 60 degree is same as the radius so
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00:29:58,559 --> 00:30:17,610
what is the purpose of stating this in fact
he says state so in fact aryabhatta is going
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00:30:17,610 --> 00:30:19,389
to
179
00:30:19,389 --> 00:30:28,619
make use of this in yet another wherein he
is going to present the sin table so this
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00:30:28,619 --> 00:30:34,039
is what the purpose of designing this can
be understood with the verse beginning with
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00:30:34,039 --> 00:30:45,269
so will come to that in a minute but before
going to be procedure which has been delineated
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00:30:45,269 --> 00:30:50,960
by aryabhatta for finding this time table
so we will discuss this very important verse
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00:30:50,960 --> 00:30:56,019
which has been quoted in various forums in
various ways and it is also been
184
00:30:56,019 --> 00:31:21,059
misinterpreted in various places so see the
first half of the verse basically presence
185
00:31:21,059 --> 00:31:41,730
the value 52832 so how does it go about so
4+100 multiply by 8 and is 1000 52000 so this
186
00:31:41,730 --> 00:31:48,000
refers to this number and what is stated in
the later half of the verse is so what are
187
00:31:48,000 --> 00:31:55,019
these numbers correspond to so this is the
number which has been employed in order to
188
00:31:55,019 --> 00:32:10,090
obtain something so is 10000 is 20000 so if
the diameter happens to be 20000
189
00:32:10,090 --> 00:32:17,590
then this number which was stated happens
to be the circumference so what is stated
190
00:32:17,590 --> 00:32:24,899
is the ration of a circumference to be diameter
the use of the word has been discussed in
191
00:32:24,899 --> 00:32:33,749
great length so why did aryabhatta use the
word you should simply said is something which
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00:32:33,749 --> 00:32:44,179
is nearby close by which can be understood
to be approximate so we will see how this
193
00:32:44,179 --> 00:32:53,269
has been analysed but before that i also wanted
to give this verse as an example of the use
194
00:32:53,269 --> 00:32:54,369
of
195
00:32:54,369 --> 00:33:00,620
and also to say that this same value has been
given by bhaskara in a slightly different
196
00:33:00,620 --> 00:33:20,210
way so so it represent 27 are famous so it
refers to 9 agni is 3 so 3 9 27 so this is
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00:33:20,210 --> 00:33:38,860
what it is so divided so divided by sum is
0 is 5 ok the word has been employed to refer
198
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to number 5 so why is
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00:33:55,039 --> 00:34:04,210
very good so the names you have got it but
then the significant is the following so arrow
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00:34:04,210 --> 00:34:09,240
all the sense will be simultaneously attack
he will lose yourself
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00:34:09,240 --> 00:34:19,320
completely so that is why he called ok so
this is fairly this is in the sense there
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00:34:19,320 --> 00:34:23,960
are various approximation which has been given
this is a far better approximations that is
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00:34:23,960 --> 00:34:34,070
what we need to understand so now i present
the discussion which has been presented by
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00:34:34,070 --> 00:34:50,369
bhaskara to you on the use of the word asana
so close to what so there are 2 3 things which
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00:34:50,369 --> 00:34:56,980
are used one is this one way of saying so
there is a which is very
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00:34:56,980 --> 00:35:04,109
very accurate so this is close to the very
accurate value this is one way of saying but
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00:35:04,109 --> 00:35:19,650
then he says is taken as 3 so this can be
close to this or close to that also so is
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something which is common to both in fact
the edition it was a problem so the is what
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00:35:29,290 --> 00:35:36,830
is found in edition so i was just breaking
my head so how this something with you try
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00:35:36,830 --> 00:35:44,610
to create a in your own mind is something
which is your imagination so is does not make
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00:35:44,610 --> 00:35:45,610
much
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00:35:45,610 --> 00:35:50,310
sense when something is hurt then you have
to imagine so that is why we should be so
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00:35:50,310 --> 00:36:02,280
this is just i believe because of this so
this is split it as that is my guess so then
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00:36:02,280 --> 00:36:12,059
he says it is true that your objection is
valid then he says general maths in which
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00:36:12,059 --> 00:36:23,680
is very important to understand so this will
be coded very frequently in this study of
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00:36:23,680 --> 00:36:32,559
philosophy all that so he says whenever there
is a sort of doubt which arises then you have
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00:36:32,559 --> 00:36:33,559
to
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00:36:33,559 --> 00:36:40,930
resort to the so how it has been analyse how
it has been understood so means so a certain
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00:36:40,930 --> 00:36:46,700
explanation which has been upward in the tradition
so from that we understand we understand certain
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00:36:46,700 --> 00:37:01,099
special meaning therefore we say edition says
that this is close to the accurate value and
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00:37:01,099 --> 00:37:14,640
therefore we will say means only this in this
context he says let not take the word could
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00:37:14,640 --> 00:37:24,180
be something which is close by residing close
by
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00:37:24,180 --> 00:37:34,621
this the different way of look at that is
it if you let us not debate on that so if
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00:37:34,621 --> 00:37:52,710
you say already a gross value is closer to
that so which means we will get a much gross
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00:37:52,710 --> 00:38:01,030
value for the circumference is going to make
some effort in specifying some values saying
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00:38:01,030 --> 00:38:05,460
that this si going to be much loser and therefore
use this so this does not make much sense
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00:38:05,460 --> 00:38:12,020
and therefore why purely logical reasoning
so we can arrive at the computer ahs to be
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00:38:12,020 --> 00:38:20,859
understood only as something which is very
close to the exact value then he asked question
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00:38:20,859 --> 00:38:29,660
so this is the similar question which nelakanta
also ask and then beautifully explain what
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00:38:29,660 --> 00:38:37,170
means by irrational and that which i think
professor srinivas will discuss the quotation
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00:38:37,170 --> 00:38:53,150
later so here bhaskara says by which you will
be able to exactly state the perimeter so
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00:38:53,150 --> 00:39:01,251
here is out of conclude the discussion on
the use of the word of asana the method
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00:39:01,251 --> 00:39:09,480
which has been described by aryabhatta to
find out the tabular or science so generally
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00:39:09,480 --> 00:39:17,270
in almost every astronomical work on mathematical
work we will see that a certain method has
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00:39:17,270 --> 00:39:24,290
been described in which typically they will
divide a quadrant into 24 parts so as shown
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00:39:24,290 --> 00:39:33,690
in the diagram so you take the circle if you
take this quadrant p24 and divide this into
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00:39:33,690 --> 00:39:41,720
24 equal parts and the points are marked there
p1 p2 p3 on so on so what is it that we are
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00:39:41,720 --> 00:39:42,720
interested in
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00:39:42,720 --> 00:39:52,520
so we are interested in the card links so
which is basically fine so the projections
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00:39:52,520 --> 00:40:01,540
which have been shown here p1 and p2 are basically
the science we consider this triangle on2p2
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00:40:01,540 --> 00:40:14,630
so p2n2 is basically sin if you consider this
as angle p2on2 is angle and the sign is pq
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00:40:14,630 --> 00:40:22,770
so these are refer to as and we denote them
as r sin of i times theta so theta is 90/24
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00:40:22,770 --> 00:40:33,039
so which is 3 degree and 45 minutes to 25
minutes here so the purpose is to determine
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00:40:33,039 --> 00:40:36,780
all the pini ok so this
245
00:40:36,780 --> 00:40:44,520
24 the last value obviously is going to be
op24 which is same as the radius of the circle
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00:40:44,520 --> 00:40:54,180
ok and once these values are known pini which
are referred to as in fact this could be precise
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00:40:54,180 --> 00:41:07,340
they should be refer to as total card length
ok as i showed here so this is half of it
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00:41:07,340 --> 00:41:18,160
which is what is sin but for the process of
convenient so people have started using jaw
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00:41:18,160 --> 00:41:30,309
itself to refer to so this understood from
the context ok if we know rsin values corresponding
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00:41:30,309 --> 00:41:31,470
to
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00:41:31,470 --> 00:41:39,799
the multiples of theta so frmo 0-90 degree
then for any intermediate value so we use
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00:41:39,799 --> 00:41:47,450
the position and usually this first order
interpellation employed and if more precise
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00:41:47,450 --> 00:41:52,819
values are required to later second order
interpolation formula has also been stated
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00:41:52,819 --> 00:42:05,940
i think maybe by sriram or myself will do
it later so what is the verse so in fact i
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00:42:05,940 --> 00:42:18,460
refers to eelier so this specification of
radius as equal to the value of cot 60 was
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00:42:18,460 --> 00:42:19,460
in connection with
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00:42:19,460 --> 00:42:47,440
this work so so this si the geometrical approach
to construction of the sin table given
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00:42:47,440 --> 00:42:58,970
by aryabhatta is the circumference ok so the
circumference of the circle refer to one fourth
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00:42:58,970 --> 00:43:28,710
so refer to one quadrant of a circle means
may you divide may you split so if you conceive
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00:43:28,710 --> 00:43:37,359
it as triangle and rectangle ok so without
the detailed commentary given by bhaskara
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00:43:37,359 --> 00:43:42,120
so this verse as follows the next verse which
has been presented by
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00:43:42,120 --> 00:43:52,039
aryabhatta which is analytic approach is going
to be almost impossible for us to understand
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00:43:52,039 --> 00:44:03,790
so the value of cos 60 was said to be r and
this value is basically 3438 so this is pretty
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00:44:03,790 --> 00:44:11,579
evident so because usually what people do
is in almost all the astronomical there but
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00:44:11,579 --> 00:44:23,590
for some so this circumference is taken to
be 21600 which is basically 360x60 so 360
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00:44:23,590 --> 00:44:32,060
degrees*60 minutes so the number of minutes
in the circumference or rather the number
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00:44:32,060 --> 00:44:33,060
of
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00:44:33,060 --> 00:44:43,849
units in a circle is taken to be 21000 etc
now that aryabhatta in his verse he has given
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00:44:43,849 --> 00:44:50,460
the ahh ration of the circumference the diameter
which is essentially 5 once you know that
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00:44:50,460 --> 00:45:03,619
then you know what radius is radius is 21600/2pi
so this was sort of stated by aryabhatta and
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00:45:03,619 --> 00:45:12,940
this will be very close to 3438 this value
in fact ahh bhaskara so gives all this value
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00:45:12,940 --> 00:45:21,970
so while explaining in this verse in detail
so how do we proceed so by geometrical construction
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00:45:21,970 --> 00:45:22,970
so all
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00:45:22,970 --> 00:45:27,230
that require is so this value initially you
know so you have to just take with 3434 it
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00:45:27,230 --> 00:45:33,390
is known so since you 21600 the value of 5
is 1 and therefore the radius is known ok
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00:45:33,390 --> 00:45:41,150
we will start with them then if you look at
carefully so since this was stated to be radius
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00:45:41,150 --> 00:45:52,700
consider this triangle obc so this is redius
so bc this angle is 30 and therefore the radius
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00:45:52,700 --> 00:46:00,359
is known bc is known so sin 30 degree is also
known so sin 30 degree is basically the 8
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00:46:00,359 --> 00:46:04,150
sin if you conceive it to the 24
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00:46:04,150 --> 00:46:11,270
part so 30 degree is going to be 8 now you
know since you know the radius you also know
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00:46:11,270 --> 00:46:19,799
sin 30 ok r/2 so this is basically r/2 this
is basically r/bc r/2 and therefore this is
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00:46:19,799 --> 00:46:30,150
also known no you consider this oabc so this
is the kind of which can think of so sin 60
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00:46:30,150 --> 00:46:40,010
is ab and is also equal to oc but you know
r you know r/2 so you do r square-r square
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00:46:40,010 --> 00:46:48,380
2 so you basically get this value oc so you
know sin 90 you know sin30 you know sin60
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00:46:48,380 --> 00:46:49,380
so what is to be
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00:46:49,380 --> 00:46:58,010
understood in general is if you know sin theta
then obviously you can get sin90-theta ok
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00:46:58,010 --> 00:47:11,720
so this si known ok then since 30 is known
so you also know verse 30 so verse is basically
288
00:47:11,720 --> 00:47:22,520
so r-rcos theta so sin 90-theta is basically
cos theta so r-rcos t in this diagram so oc
289
00:47:22,520 --> 00:47:38,109
is basically cos theta and cd is the ok so
od is r so oc is r cos theta so r-rcos theta
290
00:47:38,109 --> 00:47:46,770
is basically cd so this is also known so once
it is known so you have sin30 and verse 30
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00:47:46,770 --> 00:47:47,770
you
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00:47:47,770 --> 00:47:53,980
know rcos theta so r-rcos theta was 1 and
so what 30 is known we just do this so bd
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00:47:53,980 --> 00:48:07,380
bd if you see you know bc you know cd so now
you can in this triangle bcd you know bd also
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00:48:07,380 --> 00:48:17,369
so what is bd bd is ahh is basically the card
of 30 degree we started with card of 60 degree
295
00:48:17,369 --> 00:48:23,100
which is basically radius so now you have
come to card of 30 degree so half of it going
296
00:48:23,100 --> 00:48:31,210
to be rsin50 so you can keep on expanding
this kind of an argument and you will be able
297
00:48:31,210 --> 00:48:32,210
to construct
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00:48:32,210 --> 00:48:39,099
the entire sin table so the principle is simple
so rsin theta givs r cos theta and that gives
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00:48:39,099 --> 00:48:46,630
which is r-rcos theta and from sin theta and
vers theta you will be able to this is what
300
00:48:46,630 --> 00:48:51,799
account here we will be able to get theta/2
so once theta is known 90-theta is known and
301
00:48:51,799 --> 00:48:58,400
theta/2 is also known so with this we will
able to construct the entire sin table i am
302
00:48:58,400 --> 00:49:04,920
just going to show this how this seem to be
work so you know 8 sin so once you know8 sin
303
00:49:04,920 --> 00:49:05,920
it
304
00:49:05,920 --> 00:49:13,440
is 30 so it can get the 4 sin and the theta/2
and 90-theta so 16 sin is known from 4 we
305
00:49:13,440 --> 00:49:20,109
can go to second and on other side we can
go to 90-theta is 20th so you can follow this
306
00:49:20,109 --> 00:49:25,220
3 and this will give you the most of the sin
values and you could start with again 12 so
307
00:49:25,220 --> 00:49:28,770
45 degree so 45 degree all that going to do
is we have to just draw line gd so for 45
308
00:49:28,770 --> 00:49:37,030
degree all that you need to do is you have
to draw line dg see this gd gives you the
309
00:49:37,030 --> 00:49:38,760
card of 90 so half
310
00:49:38,760 --> 00:49:46,210
of it going to give you so this gd how do
you know so you know r so you find r as r
311
00:49:46,210 --> 00:49:54,880
root 2 r so root 2 r is known so then you
know 12 sin and then you follow the same 3
312
00:49:54,880 --> 00:50:01,339
so 6 sin so here if you go with 24 there is
not square so from 6 we get 13 or 18 and so
313
00:50:01,339 --> 00:50:08,770
on so you will see that in all the 24 sin
values can be easily obtained by this geometrical
314
00:50:08,770 --> 00:50:16,930
approach ok so with this now i stop the discussion
on aryabhattiya so we will continue in the
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00:50:16,930 --> 00:50:18,660
next lecture thank u