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like panini pingala is another great deal
in ancient india pingala is the person who
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systematized the chandahsastra he wrote the
chandahsastra with chandahsutras which give
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the theory of prosody both of vedic metres
as also of the classical sanskrit we touch
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again do not know like in the case of panini
exact date of pingala generally scholars place
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him around 300 before bc so in this talk i
will give a brief
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overview of the development of chandahsastra
then we will going to the interiority of what
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is meant by how do we scan a syllabus metre
of varvrtta in terms of eight ganas and then
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we going to the combinatorics ideas that well
developed by pingala in the last chapter of
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chandahsutra this are known as the pratyayas
so pingala discusses 6 pratyayas which are
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combinatorial techniques or combinatorial
tools for studying the sanskrit metres and
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these
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are the basic this at the 6 pratyayas in pingala
we will discuss chandahsastra again has a
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continues history ahh is a classic word is
a chapter on chandha then mathematical words
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like brahmagupta's brahmasphutasiddhanta what
are called matra british were introduced in
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discuss more in greater detail in very interesting
commentary on pingala chandahsastra the text
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that is most commonly study by students of
sanskrit is a book written around 1080 by
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kedararabhata is called
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vrttaratnakara hemacandra wrote again on is
another text the mathematics books like
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also discuss the problems related to the combinatorial
problems related to sanskrit macro environment
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touch then damodara vanibhusana in the very
interesting commentary on by narayana bhatta
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so we will straight away going to what is
the way in which sanskrit metre are understood
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the basic building blocks of studying sanskrit
meters are the characterization of syllables
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by so what is a syllable so you take anything
so srishti
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that is this consists of two now this syllable
are of 2 types laghu the short syllable guru
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the long syllable a consonant with a short
vowel is a laghu unless it is followed
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by a
consonant with long vowel is always a good
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in the end of each foot of a metre the last
syllable can be optionally taken to be good
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this is the definition obviously it not be
clear we nearly state it so let us take some
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very interesting world the invocatory works
of
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kalidasa abhijnanasakuntalam so
this is one foot of this meter it has 21 syllables
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now let us understand what are the laghu and
guru in this so first so first one is that
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should be a guru second civil now look it
should be a laghu but it is followed by a
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conjunct consonants so followed by that possible
becomes guru by again itself should be a laghu
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it is a short vowel but it is followed by
it is again followed by therefore it is a
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guru is the first laghu in this so it is very
complicated followed by a short vowel that
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is a
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laghu all these 3 are laghu so till you come
to are followed by therefore it is a guru
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but it is followed by that followed by kanchan
consonant that is a guru
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now the important thing is all the 4 pathas
have the same lughu guru structure
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that is the basic point of what is called
the now the way to read it actually we will
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see it in a minute the points where you have
to pass is also define in the definition of
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a metre eventually something called this very
beautiful meet in sanskrit languages which
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also
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used in several sound indian languages in
specification other languages also so this
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its definition itself is given in terms of
some units called ganas instead of saying
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is given by gggglgg llllllg glgglgg the way
to define this gana meter in terms of what
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are called ganas ganas are unit of 3 syllables
each with a particular structure of laghu
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guru so so the second part is very simple
number 7 so there is a there is a pause after
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each unit of 7
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after each unit of 7 syllable you should pause
so is characterised by pause that is a meter
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of 21 syllable there is a pause occur every
7 syllable and it has it has these ganas now
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what are these ganas so 3 syllable each syllable
can be laghu or guru therefore there will
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be 8 possibilities so there are these 8 ganas
and they have been given these things and
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how to remember that one way is to remember
these words laghu at the beginning meet it
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at end
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so a gana with laghu at the beginning which
means other 2 are guru a gana with laghu at
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the middle a gana with laghu at end so they
are called will have guru at the beginning
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middle and end so these are the 8 ganas in
terms of these 8 ganas all classical meet
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us are characterised and of course it is not
divisible by 3 you will say so many of these
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ganas and followed by a guru or laghu and
the beauty is this definition is also the
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meter which is
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define so this definition of the meters is
also classified in the it has marabha ma is
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ggg ra is glg bha is gll na is lll ya as you
can see is lgg then the triplet of lgg lggg
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lgg so this si the correct and so that is
the that is the way this is constituted now
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there is another very nice mnemonic to remember
the ganas this si the formula which is attributed
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to panini this is the formula which will not
find in any classical sanskrit work on prosody
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so but sighting 4 5 years ago i think donald
duck computer programming
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ahh which are he wanted to know ahh here it
is various people searched so there is a book
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on telugu prosody by a man call charles brown
written in 1843 where he has coated this but
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almost all students of sanskrit know
this is thought orally by everybody and general
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it said that this goes back to panini so what
is this in this all the ganas are encoded
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linearly characterization of the guru guru
guru characterization of jagannath that is
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lugu guru
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if you remember that you know what is and
if you write replace guru by 0 laghu by 1
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you have a binary sequence of 10 increase
if you remove the last 210 which is same as
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the beginning 10 you can put them on a circle
you have a binary sequence of length 8 now
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this binary sequence of length 8 is a special
sequence this 100 1 001011 so each triplet
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here is a binary sequence of length 3 if you
put this on circle give you put this on
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a circle you will find that it generates all
possible triplets binary triplets of length
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3 and such a cycle in today world called in
communication cycle so is the oldest example
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of such a cycle so you can have such cycles
for binary sequence of length 4 there are
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16 of them again you can put them on a circle
and generate all possible binary sequences
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of length 4 by a chord of length 16 like that
in general but as well as concern
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the ganas are 8 and we need to know now we
come to the all these ganas are defined in
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also so in 8th chapter the last chapter of
chandahsastra i think like panini ashtadhyayi
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pingala chandahsastra is also a as 8th chapter
he introduces 6 pratyayas of chandahsastra
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the first one is called prastara now had a
particular structure in terms of guru and
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laghu it has 21 syllables so at each place
you can have guru or laghu the particular
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choice has been made
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and you obtain the metres but you have to
know like a mathematician that pingala was
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so what are the possible meters with 21 syllables
can you write them down can you understand
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something about them so these are basically
dealing with these questions are called questions
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combinatorics and indians were one of the
greatest specialist in combinatorics so whenever
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they had various things the first question
is how to classify them how to put them
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in a sequence and how to understand them what
follows what so this is what pingala does
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of all this made meters so is a rule by which
you can write down all the possible metres
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of a particular length so if the length is
only 3 there are only 8 possibilities these
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are 8 gana so you have an array of 8 rows
but if have 4 syllables an array of 16 rows
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which have 5 syllable 11 array 32 rows in
obviously when you reached it will be an array
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of very very large
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number 2 to the power 31 so is the rule by
which you can write down now one you have
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written down then the question is the next
mathematical questions is suppose i tell you
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have certain role number then you tell me
what is the metrical pattern that appears
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there and suppose i give you a metrical pattern
then you tell me which row of the it belongs
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so these 2 are mathematical questions one
is called another is called uddishta it is
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called
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if you have written down the on the ground
and then wind has blown and then gone away
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but then i want to know what is 20th row in
the 5 syllabus 32 rows so i want to know what
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is that is the 20th row so is the problem
without doing the once again i should just
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go and write down what is the metrical form
what is the it appear in the prastara then
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there is which already have told you it is
2 to the power n each slot each
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level can be laghu or guru so if you have
n syllable the number of possible need is
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2 to the power n that is the then deals with
how many meters are there of a 7 syllable
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of which there are 3 gurus so how many 3 guru
meters are there in 7 syllable prastara so
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obviously this will lead us to the binomial
coefficients which we saw in the introductory
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lecture that is so now we will going to each
of them and as you can see the crucial mathematic
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that is laghu
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and guru and binary sequences so the crucial
mathematics that will appear now is binary
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mathematics and pingala was the great originator
of all the binary mathematics and this was
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of course he discovered in 1990 so first is
prastara so how do i list all is meter of
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the particular length so if you have 1 syllable
there only 2 g and l if you have 2 syllable
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we have gg lg gl and ll so pingala is giving
a rule in a very very qualified form is
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basically saying to form a pair g below l
right that below each other and fill up the
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right with 2 gs and 2 ls so pingala rule is
take the prastara of the previous put it below
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itself and in the right fill it up with equal
number of gs and ls so let us see how we get
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the prastara of 3 syllable from the prastara
2 syllables this is the prastara 2 syllable
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let us go so now this portion if you see this
is the same as the prastara 2 syllable here
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gg lg gl ll gg lg
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gl ll so this portion is prastara 2 syllabus
this portion is prastara 2 syllables in the
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right you write 4 g and 4 ls have obtained
the prastara 3 syllabus so prastara should
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have each and every metrical form of that
when should appear one and only one and it
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should be a rule by which you generate all
of them so that is the ruler of the prastara
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today the word prastara will be a mostly in
i will combinatorics i will tell you how the
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mathematical theory of prastara in music is
discussed by the rule for doing prastara this
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is a different kind of a rule supposing you
start with some row of the prastara in 4 syllabus
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and you want to know the next row without
having to start from the beginning and doing
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all that so this rule is found in so going
from one row to next row the rule is the following
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start scanning from the left once you encounter
a g put a l below bring down
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whatever it to the right as it is if there
is something else to be written in the left
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fill that up with the gs so let us do the
next row start from the left first time g
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is encounter put the l there bring down whatever
l is to the right as it is in the left fill
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it up with gs next row start from the left
below that g put an l bring down what is there
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to the right as it is start from the left
to identify the first g out the l below that
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bring down whatever is there to the right
as it is fill up
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the left with gs like that you can write this
is transforming one binary sequence to another
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in such a way that you cover all of them actually
we know something transformation of all binary
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sequences from 0 to 1 today something like
that is what you meant here so in this prastara
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you can see that are you can see that are
you can see it here so this is the four-syllable
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prastara start with gggg and then go down
there is one more way of doing this
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prastara there all the first row is the first
column is gl gl gl gl second column is gg
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ll gg ll gg ll third column is gggg llll gggg
llll fourth column is gggg lllll so that is
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what is called the ahh super fast alogorithm
of pingala to do the prastara so now we understood
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prastara this is to list all possible metrical
form of the given length now we can do a small
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thing and she is like the relation of this
with the binary numbers suppose i put wherever
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g of s0 and wherever l as i put 1 let macro
environment take the 6th row here the 6th
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row of the
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four syllable prastara suppose i want to 6th
row so the 6th row is lglg i put 1 where l
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as and the 0 the g as s i take the mirror
of this mirror image of this and you now view
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it as a ordinary binary number today during
the design you take the mirror image of this
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and understand the ordinary binary number
so as a binary number this will be +1 as a
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binary number this is 4+0+1 you have to add
1 to it because this prastara is starting
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with the first row
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with gggg is the binary number 0 so should
add 1 to it so if he said g=0 l=l is good
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to remember this to understand what everything
like that because we are familiar with binary
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numbers we do not know pingala therefore this
is a good way of and then we see that each
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metric pattern is the mirror reflection of
the binary representation of the associated
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road number -1 so this is a representation
of number 5 having understood prastara let
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us go to sankhya today we know that this number
is 2 to the power n now prastara does not
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say the
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number of metres of n syllables is 2 to the
power n he gives an algorithm for calculating
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it and the algorithm being an ancient algorithm
happens to be one of the most efficient algorithm
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for calculation the n power of a number so
the he is saying take this power so if you
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are looking for the prastara or n syllable
take me number n now start dividing by 2 if
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the number can we have you just write 2 somewhere
just put a mark 2 somewhere if the number
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cannot be half you detect 1 and instead of
putting 0 at that point you put a symbol 0
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so this is where the symbol 0 of s in pingala
it can be hard put it to 2 if you are to remove
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number 1 put mark 0 there then after you have
mark all these 2 set 0 suppose you have something
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like this start from the left wherever 2 of
the wherever wherever 0 of s multiplied by
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2 wherever 2 of s to square of the number
so ultimately to
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calculate 2 to the power n pingala is giving
you a sequence of operation which involves
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multiplying by 2 and squaring obviously 2
to the power n is multiplying 2 by itself
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n time
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but what pingala is giving you a algorithm
which is much smaller number of set so let
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us look at it with a simple example and this
is equal to 6 so 6 is divisible by 2 so you
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mark 2 3 cannot be half so subtract 1 by 3
mark 0 2 can be half you mark 1 you have
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reach subject 1 i mean mark 0 so you have
a sequence 2 0 20 and now from the right wherever
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2 appears you multiply by 2 so corresponding
to this 0 you start with the number 1 zero
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is there multiply by 2 2 appear square the
resulting thing again 0 of s multiply by 2
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2 appears square that you get to you can justify
this straight away by going back to the power
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whenever you can have it you can see you can
go back by squaring it wherever you cannot
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you can see it in also single multiplication
and that is how this algorithm is ok in fact
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pingala became the standard way of calculating
the n power of a number normally the n power
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of a number of s in simple mathematics when
we calculate the sum of a geometrical series
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so in any book like i need to sara sangraha
lilavati etc the some of the geometrical series
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is given in terms of this algorithm and so
modern scholars when they started looking
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at it they were
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00:21:09,740 --> 00:21:19,430
totally dump found it they did not know what
has been where is the geometric series and
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00:21:19,430 --> 00:21:27,320
you know you have to calculate the n power
of the ahh each factor that appears in each
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term of the geometrical sequence and here
we all have something divide by 2 but it appears
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that this actually is much faster way supposing
you are a geometric sequence and submit up
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00:21:37,920 --> 00:21:43,940
to 845 terms or something instead of 845 operations
you are something like log n instead of doing
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00:21:43,940 --> 00:21:44,940
n
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00:21:44,940 --> 00:21:47,120
operations after using the binary sequence
of the number n this reduces sequence something
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like log n of course is not the most optimal
algorithm taking the other data most powerful
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today but today need much more complex devices
so pingala then discusses ahh some few other
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00:21:59,030 --> 00:22:05,710
relations which are like some geometric series
and things like ahh that number of meters
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00:22:05,710 --> 00:22:10,350
up to n syllables then what is the relation
between the number of
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00:22:10,350 --> 00:22:14,580
meters in n+1 syllable with number m syllable
so clearly is very clearly away that 2 to
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00:22:14,580 --> 00:22:20,020
the power n but he never tells you that sn
is the number of metres of length n is 2 to
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00:22:20,020 --> 00:22:28,740
the power n in the way we would do straight
away there is another thing pingala also does
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00:22:28,740 --> 00:22:33,780
these meters that we have been discussing
are what are called that is all the 4 metre
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00:22:33,780 --> 00:22:42,650
is supposed to 4 feet 4 pathas so the kalidas
was that is at 4 padhas all the 4 padhas if
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00:22:42,650 --> 00:22:43,650
you
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00:22:43,650 --> 00:22:50,429
have the same structure of lughu guru it is
called a if the first and third has the same
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00:22:50,429 --> 00:22:59,440
structure second and fourth have the same
structure it is called and if it is not a
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00:22:59,440 --> 00:23:07,440
it will be called and so next mathematical
problem is how many of length l so pingala
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00:23:07,440 --> 00:23:17,980
in this chapter of this chandahsutra he seems
to be a very general mathematician as you
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00:23:17,980 --> 00:23:25,600
can see his discussion on mathematical questions
that arises not but
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00:23:25,600 --> 00:23:32,410
these are the sum of those were very common
causing the problem and giving a solution
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00:23:32,410 --> 00:23:43,340
of it so we know the number of length n is
2 to the power n now obviously will have n
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n n n the first 2 ns are all different therefore
it is 2n syllables but it is 2 to the power
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00:23:54,130 --> 00:24:02,250
n square if you want to understand it 2 to
the power 2n but meter are different from
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00:24:02,250 --> 00:24:07,809
each other so you should subtract the sum
of the and therefore you get the actual at
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00:24:07,809 --> 00:24:08,809
the sum of
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00:24:08,809 --> 00:24:14,140
the the number is 2 to the power n square-2
to the power n if you reflect on it for a
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00:24:14,140 --> 00:24:23,510
minute you will see how this year so where
the first line of the meter and the third
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00:24:23,510 --> 00:24:29,540
line of the meter structure second line and
fourth line are the same structure so it is
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00:24:29,540 --> 00:24:36,070
equal to having look all possibilities 2n
but we have to subtract the possibilities
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00:24:36,070 --> 00:24:45,840
with the first and second are the same and
therefore 2 to the power n squared-2 to the
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00:24:45,840 --> 00:24:47,120
power n in the same argument will lead
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00:24:47,120 --> 00:24:52,270
to the number of of n syllable to be 2nsquare-2n
whole things square-2n square-2n all these
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00:24:52,270 --> 00:25:00,140
getting having these 3 remove the number which
was query now we go to the and basically as
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00:25:00,140 --> 00:25:02,001
you can see the prastara is essentially table
of converting this pattern in two binary numbers
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00:25:02,001 --> 00:25:03,570
essentially involve knowing a number how to
find its binary representation and given a
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00:25:03,570 --> 00:25:14,870
binary representation how to know the number
essentially it
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00:25:14,870 --> 00:25:17,440
is the basic binary mathematics so pingala
has 2 rules you want to find out the metric
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00:25:17,440 --> 00:25:24,220
pattern in a particular row so row k of a
prastara of n syllables if you start with
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00:25:24,220 --> 00:25:31,510
k if it can be halved if you cannot be hard
at 1 to it halve it and go on and fill up
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00:25:31,510 --> 00:25:37,450
as many numbers as the prastara is supposed
to have and you will get the metrical pattern
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00:25:37,450 --> 00:25:43,000
so we will quickly see the example so we had
this prastara four syllables this is the prastara
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00:25:43,000 --> 00:25:44,000
four
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00:25:44,000 --> 00:25:47,390
syllabus right way metres which have 4 syllable
that is 16 in number question is being so
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00:25:47,390 --> 00:25:51,970
we are asking various question which with
respect to this what is the ahh 7 what is
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00:25:51,970 --> 00:25:58,530
8 meter and given any metre which row it appears
so it is good to remember that prastara find
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00:25:58,530 --> 00:26:03,960
the seven metrical column in the four syllable
prastara so we go back to the rule that he
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00:26:03,960 --> 00:26:07,590
stated just before so 7 cannot be have add
½ you get a guru or you can so you get a
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00:26:07,590 --> 00:26:08,590
guru
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00:26:08,590 --> 00:26:10,790
followed by laguna 2 can we have you get guru
laghu laugh now you cannot stop here because
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00:26:10,790 --> 00:26:17,020
you are looking for meters or four syllable
so do this operation once more you will get
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00:26:17,020 --> 00:26:24,820
a guru on the right so gl lg is the form which
is in the rows 7 of that prastara obviously
251
00:26:24,820 --> 00:26:31,490
you know that you to be there but just to
demonstrate that there are row 7 of that prastara
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00:26:31,490 --> 00:26:35,390
suppose if you look simple for the case 4
but prastara of like something it will be
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00:26:35,390 --> 00:26:38,780
an interesting mathematical problem now we
go to uddista so uddista is given
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00:26:38,780 --> 00:26:44,320
a metrical pattern what is the row in which
appears in the prastara so again pingala is
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00:26:44,320 --> 00:26:50,490
only 2 sutras for this you start from the
opposite direction scan the material pattern
256
00:26:50,490 --> 00:26:57,309
from the right so gl gl lg here start scan
it from right end start with number 1 whenever
257
00:26:57,309 --> 00:27:03,590
you find a laghu multiply it by 2 whenever
you find a guru you multiply by 2 and remove
258
00:27:03,590 --> 00:27:06,130
1 you multiply by 2 and remove 1 so this is
the rule of pingala start from the right so
259
00:27:06,130 --> 00:27:07,260
gl lg we have
260
00:27:07,260 --> 00:27:12,160
start with 1 that is just starting number
so whenever you beginning you go to a g nothing
261
00:27:12,160 --> 00:27:16,340
will happen 1*2-1 is 1 so you have to really
come to the first laghu that is why he saying
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00:27:16,340 --> 00:27:23,430
you can just go up to the first laghu that
you will find here so the first laghu is here
263
00:27:23,430 --> 00:27:28,970
so when we find the laghu you multiply by
2 1*2 is 2 then again you find another laghu
264
00:27:28,970 --> 00:27:35,120
you multiply that by 2 your get 4 finally
you have a guru so this 4 has to be multiplied
265
00:27:35,120 --> 00:27:37,429
by 2 and 1 subtracted
266
00:27:37,429 --> 00:27:44,039
you again come back with 7 in that this we
have seen just a minute ago that this was
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00:27:44,039 --> 00:27:46,820
the seventh metrical pattern have just given
us the process now there is a much simpler
268
00:27:46,820 --> 00:27:51,169
way calculation this is much simpler what
pingala is given is optimal algorithm
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00:27:51,169 --> 00:27:55,650
but people like us would like to know it in
relation to binary numbers be there is a much
270
00:27:55,650 --> 00:28:00,330
simpler way of doing the process so this si
described in so to do this uddisata process
271
00:28:00,330 --> 00:28:05,660
so let us take some gllgl so this is a or
you can put one more so this is a 6 level
272
00:28:05,660 --> 00:28:11,830
prastara you want to do from the left start
with 1 and keep multiplying it by 2 at each
273
00:28:11,830 --> 00:28:23,010
state and add all the numbers above l and
add 1 to it so add all these numbers so this
274
00:28:23,010 --> 00:28:29,470
is 2+4+16+32+1 so that is the row in which
this pattern will appear in the 6 syllable
275
00:28:29,470 --> 00:28:35,580
prastara so basically it is 2 to the power
1 2 square 2 cube 2 to the power 4 3 to the
276
00:28:35,580 --> 00:28:43,650
power 5 so you have 1*2 to the power 1 ahh
+ ahh 1*2 square+1* 2 to the power 4+1*2 to
277
00:28:43,650 --> 00:28:44,650
the power 5+1 and
278
00:28:44,650 --> 00:28:51,910
remember if i had written this number in terms
of if i had written this number in terms of
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00:28:51,910 --> 00:29:00,390
binary code putting 0 for g and 1 for l for
this is how this will appear and if i take
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00:29:00,390 --> 00:29:07,700
a mirror image of this while doing a mirror
image of this this is 110 110 now you will
281
00:29:07,700 --> 00:29:14,000
see this rule stated that teach place value
in binary will correspond to the power that
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00:29:14,000 --> 00:29:20,300
we are multiplayer g of s is all 0 so you
do not add only the l have to be added with
283
00:29:20,300 --> 00:29:22,520
the appropriate binary place
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00:29:22,520 --> 00:29:28,940
value and that is the rule pingala does not
want to give this rule is rule is much more
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00:29:28,940 --> 00:29:35,010
simply because you do not have to go on calculating
all this powers of 2 which you do not need
286
00:29:35,010 --> 00:29:40,309
so whenever g of s multiply by 2 and subtract
1 whenever laghu of the laghu of s is double
287
00:29:40,309 --> 00:29:45,161
and keep going that is much optimal algorithm
that he gives us for gllg i have written down
288
00:29:45,161 --> 00:29:52,090
the process from the left write 1 2 to 2 square
2 cube wherever l of s add those numbers
289
00:29:52,090 --> 00:29:56,500
that will be the so both processes of pingala
are essentially based on the fact that every
290
00:29:56,500 --> 00:30:02,429
number has a unique binary representation
that is every number can be written unique
291
00:30:02,429 --> 00:30:06,130
clear the summer powers of 2 to the power
n now we come to the last which is called
292
00:30:06,130 --> 00:30:11,210
the lugakriya so given ahh in the en syllable
prastara how many metres are there which have
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00:30:11,210 --> 00:30:14,980
5 gurus or how many metres are there which
have 7 laghus so that kind of a
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00:30:14,980 --> 00:30:20,590
question pingala has given just one sutra
at this array which is the last sutra pingala
295
00:30:20,590 --> 00:30:25,500
chandahsastra and what was the previous sutra
the previous sutra the previous sutra was
296
00:30:25,500 --> 00:30:32,360
and then that is followed with the sutra in
that the sum convention hall so that you repeat
297
00:30:32,360 --> 00:30:38,660
the sutra when you reach the end of text so
many people said that this is nothing to do
298
00:30:38,660 --> 00:30:46,120
with just the statute repetition of the sutra
in the end but the commentators edit edition
299
00:30:46,120 --> 00:30:47,120
does
300
00:30:47,120 --> 00:30:49,679
not think so ahh the commentator is very clear
that pingala is explaining how to calculate
301
00:30:49,679 --> 00:30:58,400
how to do the process with this sutra so he
is saying in that that previous i did not
302
00:30:58,400 --> 00:31:05,730
explain that sutra 8 33 was there what pingala
was saying is the following that is the previous
303
00:31:05,730 --> 00:31:09,830
which is the number of meters of all syllables
up to number n is equal to twice the number
304
00:31:09,830 --> 00:31:12,909
of metres which are length n+/1 all all metres
of length 1 2
305
00:31:12,909 --> 00:31:17,062
3 4 up to n then the number of possibilities
is twice the number of metres of length n-1
306
00:31:17,062 --> 00:31:22,429
and then the next sn+1 the number of meters
of length n+1 is just twice the previous number
307
00:31:22,429 --> 00:31:25,950
without the reduction of 1 so subtracting
1 the number of metres of length n+1 is just
308
00:31:25,950 --> 00:31:31,370
poonam the whole of double of the number of
meters of the previous length sn sn+1 is 2sn
309
00:31:31,370 --> 00:31:37,130
this for the meanings of the previous 2 sutras
of pingala but now then he comes with one
310
00:31:37,130 --> 00:31:42,270
more sutra here this will give us the so what
is we will come to this quotation in a minute
311
00:31:42,270 --> 00:31:48,020
let us just see the way how will explain says
you form this figure how do you form this
312
00:31:48,020 --> 00:31:54,059
figure first you write the square with 1 entry
below that you write 2 square below that you
313
00:31:54,059 --> 00:31:59,270
write 3 squares below that you write 4 squares
like that you go on in the first square you
314
00:31:59,270 --> 00:32:04,700
put 1 in the next 2 squares you put 1 then
below that in any square
315
00:32:04,700 --> 00:32:11,130
you put the sum of the numbers which is above
it so here you only put 1 in 2 you put both
316
00:32:11,130 --> 00:32:20,270
of them and here it is 1 so the total sum
of the total sum of the 2 cells that are above
317
00:32:20,270 --> 00:32:29,360
it is to be entered so that is the meaning
of so then in the next row 1 3 is sum of 1*3
318
00:32:29,360 --> 00:32:41,170
sum of 11 1 here 4 in sum of 1 in 3 6 is sum
of 3 and 3 4 is the sum of 3 and 1 1 and by
319
00:32:41,170 --> 00:32:50,570
now also know what these are these are nothing
but the terms which will appear when we do
320
00:32:50,570 --> 00:32:51,630
a+b to the
321
00:32:51,630 --> 00:33:02,860
power n or 1+1 to the power n these are the
numbers which appear as the various terms
322
00:33:02,860 --> 00:33:10,390
and so each of them is a binomial coefficient
mcr stands for is a number of combination
323
00:33:10,390 --> 00:33:17,410
or object n object and this rule that each
number is the sum of the two numbers above
324
00:33:17,410 --> 00:33:22,840
is essentially a recurrence relation for this
binomial coefficient it is a well-known reconciliation
325
00:33:22,840 --> 00:33:29,150
for the binomial coefficient that is the pingalas
recurrence relation which is coming in so
326
00:33:29,150 --> 00:33:37,790
we can now read this i will just to feel happy
that are you there is actually saying it written
327
00:33:37,790 --> 00:33:46,919
1 square about below that half going outside
right to space that is the square root touch
328
00:33:46,919 --> 00:33:52,280
in the middle of the first square so that
is how is to write below that right 3 squares
329
00:33:52,280 --> 00:33:56,880
below that right 4 squares till the number
of syllables that you are considering or considering
330
00:33:56,880 --> 00:34:03,610
syllables 7 syllable have to go to 7 throw
this is called f so put number 1
331
00:34:03,610 --> 00:34:06,990
in the first put into effect the following
recurrence relation so whatever number that
332
00:34:06,990 --> 00:34:12,940
has come up in the upper 2 cell you put the
sum of the whole of it in the next row that
333
00:34:12,940 --> 00:34:17,290
so then he explains what will be the first
row then what will be the second row etc etc
334
00:34:17,290 --> 00:34:25,619
that is how is explain now this si the way
the portion also studied in modern times and
335
00:34:25,619 --> 00:34:32,991
this figure has a very very famous name in
modern time it is called the
336
00:34:32,991 --> 00:34:52,039
pascal triangle slightly rotated you rotate
it slightly it is what is called pascal triangle
337
00:34:52,039 --> 00:35:09,849
pascal is a very famous mathematician wrote
a tract on this and he discuss the properties
338
00:35:09,849 --> 00:35:27,990
of all the binomial coefficient coming out
of it of course the history of this goes back
339
00:35:27,990 --> 00:35:29,549
to pingala
340
00:35:29,549 --> 00:35:40,619
but there are many many other cultures with
chinese and islami also have this pascal triangle
341
00:35:40,619 --> 00:35:50,630
much before as well obtained it is an unfortunate
thing that all our terminology refers to events
342
00:35:50,630 --> 00:35:58,380
that happened in the past 2 3 centuries and
we really are not
343
00:35:58,380 --> 00:36:21,059
aware of the contributors to development of
all our ideas so with this we come to a end
344
00:36:21,059 --> 00:36:39,250
of the study of the combinatorial techniques
that pingala started but he started really
345
00:36:39,250 --> 00:37:03,309
a host of problems now whenever once you are
set the protagonist of the study salary come
346
00:37:03,309 --> 00:37:15,469
whenever you have a list of things immediately
question is can
347
00:37:15,469 --> 00:37:41,969
i give a rule by which all this can be put
in some order and then once i put them in
348
00:37:41,969 --> 00:37:54,559
some order what place is of them can i do
is not
349
00:37:54,559 --> 00:38:00,960
a very very common problem in today's computer
science also its called permutation generation
350
00:38:00,960 --> 00:38:07,569
combination generation everywhere generation
is there you have what is called ranking unranking
351
00:38:07,569 --> 00:38:21,799
so that is such that so that sort of the virus
that pingala started his infected entire history
352
00:38:21,799 --> 00:38:32,019
of combinatorics and everywhere this question
of yours and at each place it gives rise to
353
00:38:32,019 --> 00:39:36,180
some very very interesting mathematical properties
here you saw entire
354
00:39:36,180 --> 00:40:47,709
binary arithmetic coming out of the theory
of the it is
355
00:40:47,709 --> 00:41:33,680
very interesting topic and pingala just started
and this will be seen later in prosody other
356
00:41:33,680 --> 00:43:03,670
aspects of prosody in music if you can see
357
00:43:03,670 --> 00:48:17,039
in medicine if you can see in astrology if
you can see
358
00:48:17,039 --> 00:49:34,759
in architecture in several subjects we will
discuss some
359
00:49:34,759 --> 00:50:30,459
of them later thank you
360
00:50:30,459 --> 00:51:59,269
very much