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so you would have had a wonderful overview
of indian mathematics professor srinivas so
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in this talk basically i will be covering
mathematics in vedas and sulvasutras so this
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will be run in two parts vedas as all of you
know is the most ancient stricture that is
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available to us today and sulvasutras form
a part of what are known as kalpasutras which
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itself is a part of vedas so what will be
covering is basically the
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mathematical reference that can be found in
vedas and the geometry arithmetical algebra
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etc so which can be found in the class of
text call sulvasutra the purpose of vedas
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is obviously not to deal with mathematics
so it has to do something with the relation
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of human and nature and appreciation of nature
in the form of divine been through humans
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etc so this is primarily the content of veda
and in fact it has been subsequently states
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one of the most ancient text
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called vedanga jyotisha wherein the purpose
of veda has been very clearly defined as veda
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is primarily so propitiation of the divinity
so but you might have also heard ahh perhaps
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in the overview by professor srinivas wherein
mahaviracharya so while defining the scope
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of mathematics he says to so you will see
mathematics in every aspect of life whether
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it is world later vedic or religion so any
way so what you will be trying
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to do is trace out some references which can
be found in vedas and then see what one can
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derived from those references with regard
to the knowledge of mathematics which was
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available in those ancient period so this
is primarily the idea of catching up on the
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most ancient scripture so vedic scripture
and then will want to sulvasutras so in this
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lecture we will start with the references
in vedas and then we move on discuss what
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are
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the sulvasutra text so what do they deal with
and what does the term sulvasutra mean and
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one of the most important things so as i was
mentioning veda where are you have any how
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to perform sacrifices you had to create necessary
platform and so so and the construction so
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that is involved in creating such platforms
etc is the primary content of the text called
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sulbasutras so the most important thing is
to find out the coordinal directions so that
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is one of
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the most interesting experiment it can be
done with a very very simple device called
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i will be discussing that and this is a pre-requested
for it so for doing you have to create a such
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cannot be constructed without knowing the
exact directions so that is what any cardinal
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direction when will east west north south
so everything that after all by looking at
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sunrise i will be able to determine the direction
but that is not the case because of uttarayan
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and
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dakshinayan those emotion so we need to conduct
a very simple experiment but this experiment
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is one of the most full proof experiments
so we will be discussing all that so then
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we also deal with the construction of the
geometrical objects square circle you see
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when i say both so this may be pretty simple
so oftral if i need to construct a square
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i need to draw a line and then i can have
a set square i can have a t square and then
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i i can determine what is
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the great deal about it so that is not thing
they did not have t square or set square but
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then ahh has discuss the very interesting
method by which all that you need is a couple
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of nail and then a rope so with that will
be able to construct a perfect square so that
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is what will be discussing there and then
will be moving onto what is today called pythagorean
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theorem or pythagoras theorem so we call it
sulvasutra or sulba theorem and there are
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different versions
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of both so will be discussing that and then
will also see how the triplets having listed
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in the this ancient text and then application
of sulvasutra so all that will be discussed
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in this lecture
before proceeding further so we will try to
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look into this upanishad for which is a part
of the weather so finally 10 principle upanishad
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and one of them is mundakopanishad so where
does this mathematics all within this vedic
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purpose of knowledge
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so here we have a passage mundakopanishad
so which price to classify the the body of
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knowledge itself into two groups one is called
deals with most fundamental question so who
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am i so what is the purpose of my existence
what is my connection with this all that so
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this is the domain of para and apara to all
the other things are put under para in fact
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even the vedas so the propaganda set in vedas
all under so we have 4 vedas as it is liseted
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so
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rigveda yajurveda samaveda athatveda then
we have 6 vedanga shiksha kalpana naan irukka
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jyothisham it is under this jyothisham the
mathematics also is plugged into so ganitam
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is not listed but the development of mathematics
as we have seen is so is primarily in trying
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to solve problems in astronomy and in fact
has we will see later so the connections between
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the mathematical development and the application
of astronomy will become quite evident when
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will look into the later part of a course
as to why they had to solve first order intermediate
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equation and so on in the later part of the
course so this is how ahh one is able to relate
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mathematics with the most ancient scripture
so this jyothisham ahh includes ganitham 2
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weather has said i was mentioning that you
find some references here and there and references
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will be only in the form of primarily numbers
so is primarily science of
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calculation calculating the number of days
for calculating on which day what happened
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you perform and how many number of days in
which the sacrifices to be done so there are
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various other things and how periodically
how to be done and the numbers which we find
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in the vedic literature are mostly related
to the propitiation which is done towards
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a certain particular date for instance this
this manta which you can see here to so this
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basically list all the powers of 10 starting
from 10 to the power of 2 to 10 to the power
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of 12 so refers to 10 to the power of 2 to
10 to the power of 12 so refers to 10 the
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power of 2 and towards the end of passage
we find 10 to the power 12 the list of all
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the words which refer to the power of 10 has
been listed in this similarly in the next
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one so we find to so we find all the off numbers
listed here and to
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the power multiples of 4 so these things are
listed further there is an interesting ahh
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passage in rigveda so here the passage to
so all the multiples of 10 so i am just coating
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this in order to ahh convey a certain message
that these people have been quite conversant
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with the decimal place value system so in
fact the one of the most fundamental discoveries
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so we will deal with that exclusively in a
lecture later but then you
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can see that so all the numbers which have
been mentioned for instance look at this mantra
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to so this mantra to so nava refers to 9 and
the rest 3 3 3 9 so the very interesting number
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so if you liik at so this basically some of
33 303 3003 and you get this particular sum
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and just number you also very close to ahh
the period of 18 years there have been articles
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written wherein they try to trace the origin
of this
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number and what is so special about this number
what is interesting here is to note that there
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evidence so by which will be able to see that
vedic text has been very very conversion with
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the decimal place value system and we also
have this mantra to so this shanthi mantra
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speaks of the notion of infinity to so you
take poorna poorna and what is poorna so within
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very close to the notion of infinity so now
i move on to the sulbasutra texts so i was
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mention in earlier the
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sulbasutra text gone a part of what are know
as kalpasutra in fact kalpasutra includes
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to with certain virtual which are done for
the society welfare so so has to do with the
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which was performed in the houseful so certain
text which will guide the performance of these
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things so all refer to by these things then
we have dharmasutra so dharmasutra have to
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do with the general code of contact it has
to be followed or the well
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being of the pi so that is what is and the
duties which have to be done by the people
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and so on so this is the subject matter of
durmasutra and when come to sulbasutra sulvasutras
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are takes which have been primarily composed
to assit the sulvasara in the construction
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of various sacrificial altar the first sutra
is some of this sulvasutras is basically a
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collection of bricks etc so is putting together
so putting together to create a certain platform
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on which these
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rituals can be performed what does the term
sulba mean in fact sulbasutra is a compound
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word sulba is one part and sutra is another
part sutra is yet type of writing in fact
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this has been the most ancient form of composing
text so if you look at sanskrit that is available
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to the most ancient ones have been in the
form of sutras so we have this any sutras
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you have panini sutra so so we have patanjali
yoga sutras so all of them so have
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been composed in the form of what is referred
to as sutra style suppose the definition for
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sutras so this sulbasutras is a compound word
where in the sulba as a meaning of measuring
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sulba meaning in fact the sanskrit the most
of the word are derived from the roots and
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the root is sulba sulbamanic so mana is measuring
so sulbamanae in fact this word can be derived
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ahh by a different ways in fact what is called
karmadhipati karnavati and so on so if you
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do this so sulba so this is how the word can
be derive and it refers to the act of sulab
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sulba when you say you measure something and
that entity is the object of measurement and
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that is the when you say so karana is an instrument
so by which you make measurement so all of
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them are applicable when we refer to the content
of the sulbasutra text so and therefore this
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compound sulbasutra actually means a text
which discusses the act of measuring so obviously
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the entity which is measure the instrument
which is used for measurement so all of them
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will
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be referred to into this sulvasutra text so
the sutra itself is a very very very ahh likely
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composed text so you will not find so unnecessary
words into the sutras so that is the very
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definition of sutra to and so on and the sulvasutra
was obviously ahh cannot be understood on
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their own so without that the teacher or teacher
plus commentator so if teacher is not available
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we resort to commentary which have been
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composed by later period on the sulvasutras
for understanding what does this sutra convey
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there are 7 sulbasultras so which are available
for us today these are the text both baudhayana
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apastamba katyayana manava varudha and vadhula
so of the 4 are more ahh popular and more
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studies have gone into the first 4 and the
later that have not been studied that much
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but anyway so according to scholar so baudhayana
sulbasutra seems to be the most earliest of
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sulbasutra which are available to us today
so this is based on certain analysis have
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been done
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in in the reference to the style reference
to completeness reference to the party who
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say the which is found in more text so based
on this ahh ahh felt that baudhayana must
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be the most earliest sulbasutra and one also
points to certain sutras almost similar in
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later sutra so we will find the same sutra
repeated with minor changes so and it is more
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extensive also with baudhayana sulbasutra
and therefore they feel that it is the most
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ancient one baudhayana
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sulbasutra is made up of 3 chapter consisting
of 520 sutra and composition is something
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like this so they have commentaries as they
was mentioning so for baudhayana we have 2
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commentaries one by dvarakanatha the other
by venkatesvara diksita the apastamba we have
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4 commentaries which are identified by kapardisvamin
karavindasvamin sundararaja and gopala and
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this katyayana sulbasutra we have one very
interesting
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commentary by rama or ramacandra so this rama
seems to have improved upon certain algorithms
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which have been stated in katyana sulbasutra
which i will be touching up on in my next
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lecture so we have also very interesting commentary
by mahindra or this katyana sulbasutra for
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wherein he has coated several verses some
various other things is also another interesting
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commentary now ahh i will decide a couple
of verses so which
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is found in the commentary or mahindhara so
wherein he defines what are the qualities
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that are expected of a sulbakara sulbakara
i mean the one who access so the vedic trees
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in the construction of the ah ok so where
sacrifice the perform so he says to and so
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on so means the one who is versed in arithmetic
science of calculation and so one who is verse
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with mensuration and could be of sort of enquirer
and we also
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have the adjective certain enthusiasm enthusiasm
in learning other disciplines also so today
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we have ended up into a certain situation
where in so civil engineering there will be
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some 20 disciplines so we will know only true
this even only read this so on so on so in
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the name of getting deep into something so
overall understand is all gone anyway so here
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he describes the one who should be enthusiastic
to know more and more things and so the point
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i
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want to drive through this versus is that
primarily so he is not the person who should
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be considered as the nearly geometry who is
the only that kind of a thing so this is quantities
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and there are various topics which are discussed
in sulbasutras to give you an idea as to what
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could be the content of a sulbasutra text
i have listed few of them for instance we
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have say mana is measuring is measuring the
length of the cord so which
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means is a certain technical term so they
will define those technical term but in some
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various unit of measurement are used so what
are the units of measurements linear measurements
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then 4 sided figure so construction of 4 sided
figure square rectangle and so on then here
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refers to square root ok so karani in fact
karani have various conversation i will discuss
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that little bit later so the means how do
we find the square root of a non
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square number ok so this things are discussed
there then so parinama is a transmission so
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if you have a geometrical of this particular
shape you would like to transform that object
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into some other object and this is a ahh area
of preserving confirmation so a square of
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a certain area should be transformed into
a circle of the same area then obviously you
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need to know the value of pi so in some sense
so all that so some approximations for that
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and how do you
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do geometrical so these are all discussed
in this text they are very interesting things
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and so vedi as i told you is a sort of sacrificial
ground so for different sacrifices if it comes
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at the altar in different sizes and shapes
and things are there are certain altar which
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have been constructed in the form of bird
so there is some constrain which is impose
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so it should have only one 20 square meters
so i am just saying in some units so 120 square
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meters 100 square
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meter we should construct alter in the form
of bird so which means a lot of calculation
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has to go into has to go what kind of bricks
you have to design and it has to be of certain
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height also so different layers have to be
arranged in a particular way so that the structural
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stability is also taken care so you have to
just pile the bricks so all that are discussed
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here and they also discuss about the kind
of material which has to go into preparation
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of this this is another
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interesting aspects which point in sulbasutras
regarding the construction material and the
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process so which has to be done to see that
the bricks that you create are solid enough
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and so on and so forth ahh then how to manufacture
bricks so these are all ahh the contents of
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a typical sulbasutra text so not every text
will discuss every aspect of it ok the topics
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that would discuss here is roughly so i will
start with the
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determination of cardinal direction so this
is called direction on enormous finding so
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that i will do as the first thing then how
to construct perpendicular bisector suppose
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00:22:01,390 --> 00:22:06,100
you have written in east west line then you
have to construct a perpendicular bisector
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00:22:06,100 --> 00:22:10,779
to get the what are the methods which are
employed then construction of figures like
199
00:22:10,779 --> 00:22:17,600
square trapezium circle so these are all very
interesting things as i told you ahh they
200
00:22:17,600 --> 00:22:21,899
are just done with nail and a rope
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00:22:21,899 --> 00:22:27,759
so emancipation of certain geometrical principles
for instance the sulbasutra sulba theorem
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00:22:27,759 --> 00:22:33,899
ok sulbasutra is the ancient text which has
the pythagorean theorem so in the most general
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00:22:33,899 --> 00:22:41,669
form not only templates so this principle
and how do we make use of these principles
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00:22:41,669 --> 00:22:49,549
see with pythagorean theorem which is called
in sanskrit so this is so fundamental that
205
00:22:49,549 --> 00:22:56,559
it plays a major role in trying to even transform
a particular geometrical object into another
206
00:22:56,559 --> 00:22:57,559
in shape
207
00:22:57,559 --> 00:23:03,840
and so on i will see little later ahh then
construction of citis so these are the various
208
00:23:03,840 --> 00:23:17,350
topics that we will discuss ok so determining
the east west line as you can see in the diagram
209
00:23:17,350 --> 00:23:27,070
so here ox refers to the a small stick which
is well shaped and there is a tip at the corner
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00:23:27,070 --> 00:23:35,980
ok so it is sharpened at the top end which
is ox so then you draw a circle around there
211
00:23:35,980 --> 00:23:43,479
and this sutra so basically tells you that
you have to create a certain horizontal plane
212
00:23:43,479 --> 00:23:44,479
so
213
00:23:44,479 --> 00:23:54,499
that is what is referred to by the word means
a horizontal plane so instead of fixing horizontal
214
00:23:54,499 --> 00:24:03,049
plane shanku is the stick ok so this is the
device so nikaya is having place it well then
215
00:24:03,049 --> 00:24:12,539
he says some so raju is rope okay the cord
so with which you will draw circle and all
216
00:24:12,539 --> 00:24:23,029
that you will do all measurements means so
you have to choose a rope of appropriate dimension
217
00:24:23,029 --> 00:24:27,200
which can be measured with the shank so which
means the rope has to be at least
218
00:24:27,200 --> 00:24:31,559
twice the shanku then only you can measure
with that so it is in that sense and that
219
00:24:31,559 --> 00:24:36,429
has a certain astronomical significant which
will not discuss now so if a circle that you
220
00:24:36,429 --> 00:24:41,499
draw is too small then while doing a shadow
measurements so the shadow need not even enter
221
00:24:41,499 --> 00:24:46,549
the circle here so what you will do it so
you create a platform you place the stick
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00:24:46,549 --> 00:24:52,960
and then you see the shadow so the shadow
so the link keeps on decreasing and at a certain
223
00:24:52,960 --> 00:24:54,009
point it will enter into
224
00:24:54,009 --> 00:24:58,720
the circle and then in the afternoon it will
move out of the circle so these two points
225
00:24:58,720 --> 00:25:06,940
to be primarily mark and they are a and b
in this figure points a and b so so finally
226
00:25:06,940 --> 00:25:15,259
define the east west and the east point ok
so once you draw a line a b so a is marked
227
00:25:15,259 --> 00:25:21,240
in the forenoon as the shadow enters into
the circle and b is marked as the shadow exist
228
00:25:21,240 --> 00:25:25,340
from the circle so this experiment can be
performed anywhere so the only thing that
229
00:25:25,340 --> 00:25:27,419
you have to ensure is a at
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00:25:27,419 --> 00:25:34,899
surface in fact my friend professor srinivas
so once he was mentioning that his friend
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00:25:34,899 --> 00:25:43,179
so consulted somebody ahh regarding vasthu
so then the person came up and said your house
232
00:25:43,179 --> 00:25:48,999
is not properly oriented in the east west
direction he was married once so this fellow
233
00:25:48,999 --> 00:25:57,159
said what i do and so on so then apparently
he asked how did that person who came for
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00:25:57,159 --> 00:26:03,879
serving tell you that your house is not properly
oriented apparently he said he brought a small
235
00:26:03,879 --> 00:26:04,879
device where in
236
00:26:04,879 --> 00:26:12,119
the magnetic needle was there and then so
he said that so this is not properly oriented
237
00:26:12,119 --> 00:26:18,299
so this magnetic needle so how big size it
is when device can different magnetic your
238
00:26:18,299 --> 00:26:23,690
own card perhaps so can deect the magnetic
needle and that will not give you the exact
239
00:26:23,690 --> 00:26:29,009
is which direction so in fact the card of
full proof method is this method so it where
240
00:26:29,009 --> 00:26:35,650
in there are so many devices and this is not
something so this is so reliation so this
241
00:26:35,650 --> 00:26:38,470
is a very simple
242
00:26:38,470 --> 00:26:44,369
technique will be able to see and later when
we saw that this is exactly oriented any way
243
00:26:44,369 --> 00:26:48,509
the point i am trying to convey here is the
very very experiment and full proof experiment
244
00:26:48,509 --> 00:26:54,340
which can be done at any place on any given
day so which will give you the exact east
245
00:26:54,340 --> 00:27:03,889
west direction ok so this is a slide so which
has been prepared based on certain work which
246
00:27:03,889 --> 00:27:10,859
is given in an astronomical work called so
with myself and profession sriram
247
00:27:10,859 --> 00:27:15,470
was studying at some point of time so it is
a very interesting of formula which is ahh
248
00:27:15,470 --> 00:27:21,269
given here so if you look at this is the formula
it is a lot of complicated complex formula
249
00:27:21,269 --> 00:27:26,320
which involved so lot of trigonometric functions
and the inverse of trigonometry function in
250
00:27:26,320 --> 00:27:31,789
the left hand side what you find is t so t
basically refers to the time and in the right
251
00:27:31,789 --> 00:27:36,840
hand side what you find is pi and delta so
pi refers to the latitude of place and delta
252
00:27:36,840 --> 00:27:38,169
the definition of the sun
253
00:27:38,169 --> 00:27:43,000
so what i am trying to say here is so with
this simple device shanku you will be able
254
00:27:43,000 --> 00:27:50,450
to precisely tell what the time is provided
you are able to see the where the sun is this
255
00:27:50,450 --> 00:27:56,389
any distance of the sun which is he said here
once it is known will be able to get the time
256
00:27:56,389 --> 00:28:02,539
so this device is the very very profound device
so once the latitude the latitude itself can
257
00:28:02,539 --> 00:28:08,639
we of course found using this shanku at any
given place what is the latitude to the place
258
00:28:08,639 --> 00:28:09,639
simple
259
00:28:09,639 --> 00:28:16,159
experiment will tell you so this formula is
primarily convey so to give you a certain
260
00:28:16,159 --> 00:28:22,979
idea as to how even profound things can be
done with this very simple device and this
261
00:28:22,979 --> 00:28:28,059
also tells you the kind of sophistication
there gone in so as we was mentioning so one
262
00:28:28,059 --> 00:28:33,830
of the comment that was very clearly explain
as to why you need to conduct this experiment
263
00:28:33,830 --> 00:28:42,119
to determine east west so this mahidhara says
to rising point so if
264
00:28:42,119 --> 00:28:58,590
you look at these eastern part of the risen
so the rising part of the sun keep on 15 very
265
00:28:58,590 --> 00:29:05,529
very small amount every day and cannot be
known by simply looking at the eastern part
266
00:29:05,529 --> 00:29:16,519
of risen so finally he concludes to so it
is difficult to simply determine by looking
267
00:29:16,519 --> 00:29:24,020
at the sun so therefore you need to conduct
this experiment so having done this experiment
268
00:29:24,020 --> 00:29:28,840
ahh so you have to find out the perpendicular
direction that is the next
269
00:29:28,840 --> 00:29:38,849
problem so this is actually done in a couple
of ways one is which is folding of the cord
270
00:29:38,849 --> 00:29:43,240
see you have drawn a line now you have determine
the east west now you have to draw a perpendicular
271
00:29:43,240 --> 00:29:52,659
so all that you have is nail and rope nowadays
what we do is primarily this latter method
272
00:29:52,659 --> 00:30:02,200
by drawing fish figure so i will show this
one by one see the first by cord folding is
273
00:30:02,200 --> 00:30:04,619
as simple as that suppose you have determine
274
00:30:04,619 --> 00:30:13,820
point a and b all that this is you have to
take care rope which is twice a b and then
275
00:30:13,820 --> 00:30:18,100
you have to mark the centre of this it by
folding it you will be able to mark the centre
276
00:30:18,100 --> 00:30:22,759
of this rope so once you that all that i need
to do is you have to full so tie at the end
277
00:30:22,759 --> 00:30:27,680
and then pull it on both the size so once
i did not give you not others will give you
278
00:30:27,680 --> 00:30:34,700
so this is this will give you this perpendicular
to the east west line so this is the first
279
00:30:34,700 --> 00:30:35,789
method the other
280
00:30:35,789 --> 00:30:40,859
method is which is followed by even if you
ask any school student all that you will do
281
00:30:40,859 --> 00:30:45,570
is will take a campus and then you will draw
a semicircle and then put on other side draw
282
00:30:45,570 --> 00:30:50,441
the other circle so this creates a figure
see so a figure like this so this is like
283
00:30:50,441 --> 00:30:58,289
a fish and therefore it is called means fish
so by drawing fish figure we will be able
284
00:30:58,289 --> 00:30:59,469
to get the perpendicular there
285
00:30:59,469 --> 00:31:05,840
now i come to a very instructing construction
which has been discuss in baudhayana sulbasutra
286
00:31:05,840 --> 00:31:21,249
for constructing a
square see so he says so whatever be the dimension
287
00:31:21,249 --> 00:31:32,509
the side of the square ok so he says you take
a rope of that link so he says so in this
288
00:31:32,509 --> 00:31:39,519
figure p and q refers to the tip of the rope
and p q basically gives you the side of the
289
00:31:39,519 --> 00:31:47,659
square so what is to be done is so mark the
midpoint of it and you have to fix it fix
290
00:31:47,659 --> 00:31:56,889
a nail then draw a circle with a so you just
in a next figure so o is the nail and you
291
00:31:56,889 --> 00:31:59,039
draw circle so having
292
00:31:59,039 --> 00:32:05,359
done this say all that we employee is a couple
of nails and then rope that is all in trying
293
00:32:05,359 --> 00:32:19,559
to do draw a square i'm so he says means you
ahh at the centre fix the nail to to draw
294
00:32:19,559 --> 00:32:31,789
a circle then which will encounter refers
to the diameter of the circle refers to the
295
00:32:31,789 --> 00:32:46,409
diameter p q is the diameter here to see you
place a nail at t you place a nail at q then
296
00:32:46,409 --> 00:32:50,929
he says to so it is very
297
00:32:50,929 --> 00:32:58,299
special you say in fact there is one of the
ahh word which one can find in the ancient
298
00:32:58,299 --> 00:33:05,859
literature so is normally sort of removing
but pretty is a word when the suggest to go
299
00:33:05,859 --> 00:33:13,530
which is tying ok tying latching so then he
says you tie it and then the same string is
300
00:33:13,530 --> 00:33:24,609
used and you have to draw a circle ok similarly
you tie it and then draw another circle ok
301
00:33:24,609 --> 00:33:31,639
so to wherever these 2 circle interface so
which
302
00:33:31,639 --> 00:33:41,499
is e and f so these are 2 points with them
you draw a line then you have to mark points
303
00:33:41,499 --> 00:33:55,029
r and s ok to again see we fix a nail here
we fix a nail here then you say you fix 2
304
00:33:55,029 --> 00:34:00,200
more nails here and then you call got the
square how do you do so then starting from
305
00:34:00,200 --> 00:34:10,190
p you have to draw a circle it was done then
you draw a circle then you draw a circle then
306
00:34:10,190 --> 00:34:13,889
you draw a circle to once this 4 circles are
drawn so this
307
00:34:13,889 --> 00:34:18,889
doted circle all that is done is where are
those 2 circles interact intersect you have
308
00:34:18,889 --> 00:34:23,630
to mark the point and you will get the square
a b c d this is a very beautiful construction
309
00:34:23,630 --> 00:34:29,840
which has been discussed in bodhayana sulbasutra
to draw a square whatever be the dimension
310
00:34:29,840 --> 00:34:43,450
ok as long as the long as biggest model then
we have this so called pythagorean theorem
311
00:34:43,450 --> 00:34:52,570
discussed here and it is a split statement
to
312
00:34:52,570 --> 00:35:04,800
this is the statement of pythagorean theorem
ok so as i said refers to a 4 sided figure
313
00:35:04,800 --> 00:35:23,510
ok to refers to a rectangle raju is rope or
line is not to be found in the classical literature
314
00:35:23,510 --> 00:35:28,060
it can be found in the vedic literature and
in fact there are various instances in which
315
00:35:28,060 --> 00:35:33,520
the word has been used so this have been defined
by the commentator of the sulbakara also in
316
00:35:33,520 --> 00:35:40,480
fact ahh that today i incidentally saw in
chandogya
317
00:35:40,480 --> 00:35:46,220
upanishad the word has been used and shankara
in his commentary very beautifully define
318
00:35:46,220 --> 00:36:06,840
its to if you want to say for instance if
you draw a rectangle for this is a corner
319
00:36:06,840 --> 00:36:24,490
corner means this particular thing is what
is referred to us refers to this to so this
320
00:36:24,490 --> 00:36:34,180
can be taken as and this can be taken as so
all that is said here is if you think of drawing
321
00:36:34,180 --> 00:36:36,430
a square here and then
322
00:36:36,430 --> 00:36:44,610
drawing another square here if you draw a
square and this will give you the area of
323
00:36:44,610 --> 00:36:56,869
these two so
to this is something which is very well known
324
00:36:56,869 --> 00:37:02,740
so whatever be the area that is created by
the 2 sides so the hypotenuse also gives that
325
00:37:02,740 --> 00:37:13,000
so this si basically the pythagorean theorem
version as i was mentioning earlier the katyayana
326
00:37:13,000 --> 00:37:22,490
version is primarily the bodhayana sutra as
it is with the only thing added can
327
00:37:22,490 --> 00:37:27,650
be understood in a 2 different ways so i would
say that this is the most fundamental theorem
328
00:37:27,650 --> 00:37:34,110
which is to be known in this plane or geometry
it cannot be dispense with and impact the
329
00:37:34,110 --> 00:37:40,460
application or many many many food so even
in very complex calculations like the period
330
00:37:40,460 --> 00:37:47,730
of eclipse etc so all that we use is primarily
this is not a right angle triangle and so
331
00:37:47,730 --> 00:37:58,070
let see in fact the comment to it is a very
very clear explanation area so it
332
00:37:58,070 --> 00:38:02,930
consider one side whatever be the area created
by that side whatever be the area created
333
00:38:02,930 --> 00:38:11,420
so that particular thing creates the area
so so this what is the meaning of the sutra
334
00:38:11,420 --> 00:38:30,280
ok this manava sulbasutra also states this
theorem so it says to this is a
335
00:38:30,280 --> 00:38:38,160
very very clear statement so ayama and vistara
they refer to the length and breadth of this
336
00:38:38,160 --> 00:38:44,990
rectangle so we multiply ayama by ayama vistara
by vistara so which
337
00:38:44,990 --> 00:38:51,080
means so if you consider one side of a find
a square and vistara find b square means adding
338
00:38:51,080 --> 00:38:59,790
them together see vargamulam is square root
of a square b square is c square that kind
339
00:38:59,790 --> 00:39:05,550
of a square root and vargamulam is square
root so that gives you so this is what essentially
340
00:39:05,550 --> 00:39:19,210
lead is ayama square+vistara square square
root is karna so soon after giving the theorem
341
00:39:19,210 --> 00:39:30,870
to gives several triplets so here is the sutra
342
00:39:30,870 --> 00:39:48,620
to so one triplets is 3 4 5 ok then is 12
12 12 5 then 15 8 so 17 square then 7 and
343
00:39:48,620 --> 00:40:06,490
24 so it is 25 square 12 35 is 37 so that
it says he says we can go on constructing
344
00:40:06,490 --> 00:40:12,970
such picture so uses a few triplet as examples
of the theorem which he stated in the most
345
00:40:12,970 --> 00:40:20,160
general form in the previous sutra sutra gives
couple of other triplets so i am just listing
346
00:40:20,160 --> 00:40:26,160
them and from al the question as to why he
chose only this triplet so many
347
00:40:26,160 --> 00:40:31,440
triplets which can be constructed so one can
think of the that this sulbasutras sulbakara
348
00:40:31,440 --> 00:40:40,830
thought of a certain general formula and plugging
in certain values will be able to get those
349
00:40:40,830 --> 00:40:51,860
triplets so that is what i show you in the
next couple of minutes or instance ahh if
350
00:40:51,860 --> 00:40:58,190
we look at one of the relations which has
been given in katyayana sulbasutra so this
351
00:40:58,190 --> 00:41:08,940
is a algebric relation na square so can we
written in this particular form so fine so
352
00:41:08,940 --> 00:41:09,940
katyayana
353
00:41:09,940 --> 00:41:15,150
sulbasutra does not presented in this particular
form but in a certain context so in finding
354
00:41:15,150 --> 00:41:19,750
the value of root and you give a to find a
certain description which can be written in
355
00:41:19,750 --> 00:41:32,070
this particular form ok datta to stating the
study of indian mathematics contribution of
356
00:41:32,070 --> 00:41:37,820
india so it is a very great scholar you have
written a text called a since of sulba so
357
00:41:37,820 --> 00:41:44,410
wherein i mean he has pi to analyse as to
what would be the rational in trying to
358
00:41:44,410 --> 00:41:52,910
present these examples theorem was stated
so this if we make a substitution n=m square
359
00:41:52,910 --> 00:41:59,100
and we plug in a=1 so then this transforms
into this particular form so if you plug in
360
00:41:59,100 --> 00:42:10,890
the value m=3 5 7 so immediately you get these
triplets ok so 3 4 5 5 12 13 7 4 and 25 so
361
00:42:10,890 --> 00:42:18,820
if you rewrite this equation in this particular
form and then you substitute the value 2 4
362
00:42:18,820 --> 00:42:24,570
6 it immediately gives you 3 more triplets
so in this you can see that so this 2 triplets
363
00:42:24,570 --> 00:42:26,140
along the same but other
364
00:42:26,140 --> 00:42:34,990
are different ok the other example which has
been given by baudhayana for instance 15 26
365
00:42:34,990 --> 00:42:45,000
39 how did baudhayana get so here the principle
behind generating the rational triangle so
366
00:42:45,000 --> 00:42:55,770
this si another way of looking at it so here
ahh there is a verdi which is called saumuki-vedi
367
00:42:55,770 --> 00:43:00,190
and in the saumuki-vedi what should be the
length what should be the it is not actually
368
00:43:00,190 --> 00:43:05,760
ah rectangle but it is quietly different kind
of thing so wherein it is a sort of
369
00:43:05,760 --> 00:43:11,610
trapezium kind of a thing so wherein so the
length of the size and so on are specified
370
00:43:11,610 --> 00:43:45,600
in that context so we
kind these suthras to
371
00:43:45,600 --> 00:43:56,120
in fact so this can eb sort of written like
this multiplication ok so you take 3 and then
372
00:43:56,120 --> 00:44:03,920
do this process so then you will get 12 16
and 20 so if you take 4 and then do this process
373
00:44:03,920 --> 00:44:24,040
then you will get 15 20 and 25 then he says
to 15 36 and 39 so this is the kind of triplet
374
00:44:24,040 --> 00:44:26,030
as i was
375
00:44:26,030 --> 00:44:33,510
mentioning so this has been given by bodhayana
also any way the principle that seem to have
376
00:44:33,510 --> 00:44:44,420
employed here in arriving at this triplet
one of this speculation is the that they had
377
00:44:44,420 --> 00:44:52,160
used general formula and then substituted
different that you see 2 3 4 5 6 so you get
378
00:44:52,160 --> 00:44:59,630
all these triplets that are found and given
by bodhayana and later by other because you
379
00:44:59,630 --> 00:45:06,850
get various in triplets and all these measures
can be ahh taken to the measure of the
380
00:45:06,850 --> 00:45:12,740
various vedis so which they have used for
various sacrifices so there is an interesting
381
00:45:12,740 --> 00:45:26,340
application of this sulva theorem so i will
discuss one application now so putting together
382
00:45:26,340 --> 00:45:39,150
which means you have 2 square of different
size and you want to create a square whose
383
00:45:39,150 --> 00:45:47,880
area will be the sum of these squares ok so
it is geometrical construction all that you
384
00:45:47,880 --> 00:45:58,020
need to do is closely follow this figure so
a b c d is one square and c g h i is another
385
00:45:58,020 --> 00:45:59,270
square so these
386
00:45:59,270 --> 00:46:14,620
are the 2 squares and now you want to construct
a square whose area is sum
387
00:46:14,620 --> 00:46:31,350
of these 2 squares to by the side of the smaller
square ok the one which is larger in dimension
388
00:46:31,350 --> 00:46:40,010
so all that he says is you have to take the
dimension c g and then mark at in the larger
389
00:46:40,010 --> 00:46:55,260
square which is same as d e ok here the term
refers to a rectangle ok so it is not it is
390
00:46:55,260 --> 00:47:04,450
so so the diagonal for this which is a e e
of this square
391
00:47:04,450 --> 00:47:18,820
which you want to construct as a sum of the
area of these 2 square will give the side
392
00:47:18,820 --> 00:47:27,200
so basically what you will get is a a h e
so this is the square ok so with this so i
393
00:47:27,200 --> 00:47:34,790
will conclude my lecture today and will have
the next part of vedas and sulbasutra in the
394
00:47:34,790 --> 00:47:43,830
next lecture thank you yes sir yeah you see
you told how to find out these correct correct
395
00:47:43,830 --> 00:47:52,810
yeah similarly to find out the time yeah there
is a smallest yoga yeah could you tell me
396
00:47:52,810 --> 00:47:56,660
that we also only calculation
397
00:47:56,660 --> 00:48:02,190
ahh from the shadow using the shadow yeah
to find out the time no no see what i said
398
00:48:02,190 --> 00:48:07,310
was this formula is encoded in the form of
a word that we have decoded in the form of
399
00:48:07,310 --> 00:48:17,960
modern notation and that what it translates
to is what i said this is a very very effective
400
00:48:17,960 --> 00:48:26,350
device to determine so many things so in fact
there is a factor called there almost always
401
00:48:26,350 --> 00:48:33,390
feels fine and the device which forms the
basis of this chapter to conduct various experiments
402
00:48:33,390 --> 00:48:34,390
and
403
00:48:34,390 --> 00:48:38,650
determine various quantities of astronomical
where is the shanku they will place it and
404
00:48:38,650 --> 00:48:43,800
then so based on the shadow will be able to
determine the lattice of it in fact means
405
00:48:43,800 --> 00:48:55,809
3 questions to you means you want to know
where you are so where you are essentially
406
00:48:55,809 --> 00:49:01,170
is given by latitude and longitude and in
fact latitude is most important thing latitude
407
00:49:01,170 --> 00:49:08,190
basically tells you how the time will uctuate
see how much duration of daylight will
408
00:49:08,190 --> 00:49:14,480
be available for you to do the gate way transaction
so this slogas of course we can go back and
409
00:49:14,480 --> 00:49:21,420
then tell you but this is not a sloga where
it is sloga is basically gives the description
410
00:49:21,420 --> 00:49:25,600
of the terms in the formula so what is the
multiply by what what is to be divided by
411
00:49:25,600 --> 00:49:34,210
1 my question is by the same argument that
the shadow of the no one should uctuate on
412
00:49:34,210 --> 00:49:43,440
day to day basis see i will tell you so namely
it will sounded to you that way but the point
413
00:49:43,440 --> 00:49:45,560
a this
414
00:49:45,560 --> 00:49:53,880
variation in the raising point is due to change
in declination of the sun the eclipse and
415
00:49:53,880 --> 00:50:01,400
equator are inclined so above 23 degrees so
from the east point sun will move to 23 degree
416
00:50:01,400 --> 00:50:09,670
towards the north return back and then south
so if you know the day on which will be going
417
00:50:09,670 --> 00:50:16,740
to be exactly saying the east point and then
it may so this will happen on the ld but suppose
418
00:50:16,740 --> 00:50:24,030
you have to do ahh the construction on any
given day at any given place so this will
419
00:50:24,030 --> 00:50:25,030
be the most
420
00:50:25,030 --> 00:50:34,230
reliable method the point is on a given day
so the path traced by the sun will be almost
421
00:50:34,230 --> 00:50:40,650
parallel to the equator so leaving out the
very minor change in the declination so within
422
00:50:40,650 --> 00:50:46,450
a few hours in which you take the shadow measurement
so the path will be so this is this path is
423
00:50:46,450 --> 00:50:52,000
apparent path which is due to the rotation
of the earth and therefore the path that is
424
00:50:52,000 --> 00:50:58,670
placed by the sun in the background so will
be more or less parallel to the equator and
425
00:50:58,670 --> 00:50:59,670
therefore when
426
00:50:59,670 --> 00:51:05,730
say does not matter how far it is moved from
the equinoctial point where will be the rise
427
00:51:05,730 --> 00:51:10,380
so there will be a corresponding shift in
this side also so and therefore the points
428
00:51:10,380 --> 00:51:16,620
that you will get by this ahh by this will
be so the line that you get will be parallel
429
00:51:16,620 --> 00:51:23,490
to the east west line so by connecting these
two points so but looking at that so if you
430
00:51:23,490 --> 00:51:27,230
say this is a rising point that rising point
will be shifted from the equinoctial point
431
00:51:27,230 --> 00:51:32,260
it is depending upon the declination the declination
will be 0 to the exact these points so that
432
00:51:32,260 --> 00:51:32,510
you know