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good morning this is the first lecture of
this course ahh which is being given on mathematics
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in india from vedic period to modern times
it is a novel course which tries to trace
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the way mathematics developed in india ahh
the first talk is an overview talk in this
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i will try to highlight those periods is there
was a significant development of mathematics
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in india i will also try to summarise ahh
the special nature of athletics as a developed
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in india
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i would like to emphasize the algorithmic
way in which most problems in mathematics
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was considered in the indian tradition so
i am flashing the outline not going to read
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it out ahh we can just see the kind of topics
we are going to follow up ahh we will cover
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the development of indian mathematics in the
ancient period indicate some highlights during
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that period then the early classical period
say 500 bce to 500 ce which culminated in
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the birth
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of aryabhatta then the development mathematics
in the latest classical period from 500 in
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the common era to 1250 we will then go to
some uses of the highlights of what happened
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during the mediaeval period till about 1850
towards then we will discuss something about
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the nature of mathematics in india how mathematics
how was results like that and then about the
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contemporary period we will speak a little
bit about the srinivasa
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ramanujan finaly know that history so some
find out pervasive mathematics in india is
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this following statement from the ganita sara
sangraha of mahaviracharya if along six packs
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statement again when mahavirachary it got
by saying that mathematics is important in
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all areas when finally concludes by saying
to that is not provided by mathematics so
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that is kind of statement that mahaviracharya
is beating weather is it in astronomy or beat
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in architecture or beat in conjunction of
granite
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position kind course of moon logic quite the
grammar so it says all purpose statement of
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mahaviracharya that mathematics provides all
aspects all subject this quotation from the
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9 century world called ganita sara sangraha
or mahaviracarya so ganita stands for calculation
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competition to is a statement due to ganesa
daivajna is a commentator of lilavati and
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therefore we can expect that
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indian mathematical text really abound in
rules to describe systematic and efficient
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procedure for calculation just to give you
an example we will go to a very ancient rule
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this is given by bhaskara i this is in this
commentary to aryabhatiya it is just a rule
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for calculating the square of a number to
so we can see the kind of calculation was
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talking about to take any number 125 first
you square the last number multiply the
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other numbers by 2 and the last number so
you get this row then move away remove one
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digit square the next number multiply by 2
and that number the next row and finally square
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the last number remove one number add all
of them the important thing to realises even
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this very ancient rule written in 1619 ad
actually uses n*n-1/2 multiplication to calculate
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the square of a number a n digit number multiplied
by another n digit number we will n square
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multiplication but since we are squaring the
same number is 2ab like that comes in and
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so you are having an optimal algorithm for
square this is the indian mathematicians always
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right to give the best possible way of the
best possible procedure for doing a calculation
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now the algorithm itself as a history ahh
it was the name given to the indian methods
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of doing calculation when we coordinates when
the name of central asia mathematician call
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who in the ninth century wrote a book on indian
methods of calculation that is methods of
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calculation using the decimal place value
system and that book was called algorithm
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latin version of that book is available the
original arabic version is not available and
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this was the book that introduced the decimal
place value notation to the arabic world and
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later on to the european world and so the
people who followed this was calculation were
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called algorithm and the algorithm comes from
() and this algorithm factor is not something
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very specific to mathematics impact it provides
all indian sciences most of indian disciplines
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sastras as we call them they do not present
a series of propositions they normally give
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you a set of rules a set of procedure which
tell you how to systematically accomplish
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something so the rules given in sastras are
usually called as vidhi kriya prakriya
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sadhana parikarma karana these are the names
and these rules are what are usually formulated
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as sutras so disciplines the ethical disciplines
in india they provide systematic rules of
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procedure rather than a set of propositions
and the for the most canonical such systematic
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text in india is the great grammar written
by panini called astadhyayi in fact most other
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disciplines and especially mathematics is
extensively influenced by the method of
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panini please use symbolic and technical devices
recursive and generative formalism and this
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system of convention that govern rule application
and rule interaction all these go back to
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panini and it has deep influence you are not
the modern discipline of linked list in fact
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many scholar actually acknowledge that place
panini holds in indian tradition plays taht
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something analogous to the place you could
hold in the ecuclidean
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tradition and here is a quotation from stall
where he saying panini is also dividing systematically
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sanskrit occurrences from a set of rules and
euclid is also deriving a set of propositions
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from a collection of axioms but the world
deriving will have 2 different mean means
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in this 2 context panini is actually generating
valid occurrences of sanskrit is not proving
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theorems euclid is demonstration is proving
theorems in mathematics from a set of
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postulates so in ancient period the ugliest
text in mathematics available are the text
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on construction of higher alters the vedis
these are the sulvasutras these are the oldest
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texts of geometry in teh world they give procedures
for construction in transformation of geometrical
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figures then there are ancient astronomical
siddhantas which deal with astronomy when
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we come to the classical period starting from
panini we then have the chandahsutra of
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pingala which initiated combinatorics we have
some mathematics in the jaina tradition in
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the jaina then more crucially the idea of
0 and decimal place value system developed
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in this period and all these terminated in
the mathematics and astronomy that is found
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in the text aryabhatiya aryabhatta which was
written in 499 of the common error most of
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the standard procedure in arithmetic algebra
geometry trigonometry were perfected and
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many more things which was used in astronomy
like the indeterminate equations sign tables
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all these things were perfected by the time
of aryabhatta so the ancient sulvasutras deal
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with lot of things units of measurement marking
directions construction of rectangle square
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trapezium transformation of square and it
has the first oldest statement of geometry
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the theorem which we attribute phithogorous
commonly it is called the bhuja-koti-karna-nyaya
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in
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later indian mathematical text it is the sum
of the two sides of a rectangle the square
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sum of the squares of two sides of a rectangle
is equal to the square of the diagonal this
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is the rule i stated in baudhayana sulvasutras
to there are even more complicated rules of
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adding squares then there is a rule for approximate
conversion of square into a circle which leads
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to a value of 5 around 3 08 then there is
a very
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interesting formula for square root of 2 is
called dvikarana in it is accurate of 2 several
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decimal places that you can see finally all
this geometry is used in constructing all
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types this is the rule in katyana sulvasutra
the problem is how to construct a square which
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is n times ahh the area of a given square
and katyana sulvasutra gives a very interesting
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geometrical formula n+1a/2 whole squared-n-1a/2
whole square is na square it is
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using this very interesting algebra it result
to calculate the side of a square which is
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n times in area of the given square pingala
sutra are the combinatorics and these are
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a very interesting diagram known as the meru
prastara which appears in pinglas it gives
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you the what we now call as the binomial coefficients
ncr they arise very naturally when you want
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to count how many metres are there which are
n syllables but in which are number of
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groups appears that is ncr as we shall see
later the decimal place value system arose
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in the ancient period the main thing about
the decimal place value system is that is
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an essential in algebraic concept the number
52038 written as 5 times 10 cube 2 times 10
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square and 0 times 10+3 is something i think
to a algebraic polynomial 5x cube 2z+square+
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0x+3 it is this algebraic future of place
value system that enabled the indian mathematicians
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to give systematic and very interesting procedure
for making calculations and they became the
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standard methods of calculation all the world
over sometimes the indian books do give some
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special techniques also which are essentially
originating out of the place value systems
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for instance the forming the buddhivilasini
of ganesa daivajna lilavati it discusses what
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is currently popularly known as the vajrabhyasa
method of multiplication vertical and cross
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wise method multiplication the history of
decimal place value system goes back to the
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vedas they use the system to the base 10 very
naturally the upanisads talk of zero and infinity
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panini's astadhyayi has a notion of ahh lopa
which is i think what is called as this idea
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of in bauddha philosophy the idea of abhava
in the naiyayika philosophy pingala's chandahsustra
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uses a 0 as a marker which not a clear whether
at that time the idea of
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0 as a number was no now soon enough the idea
of place value system became so common that
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philosophical works ahh such as vasumitras
with this text and even and yogasutra started
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explaining the speciality of the place value
system there is a quotation from the vyasabhasya
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on yogasutra to just as nad is understood
as a mother daughter-in-law or a sister ahh
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li which appears at different places that
is number 1 which
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appears at different places will have different
values and they got 10 so like this this issue
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became well known in the circles of philosophy
also and got discussed and one of the oldest
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place value system explicitly is in a book
called vrddhayanajataka written by sphujidhvaja
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around 270 in the common era and by the time
of aryabhatta aryabhatiya all calculations
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are formulated with the place value system
her an inscription in gwalior which
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is giving the number 270 it means it is appearing
here and the many other inscriptions in southeast
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asia in gwalior in various other places around
early 7 century ahh which give numbers in
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the place value system with 0 also now this
indian place value system acclaimed universally
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this statement in the 7th century by a syrians
who is this out of saying that the greek seem
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to think too much of themselves but they really
do not know the
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basic methods of calculation that indians
have discovered and they better know that
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the others also know something of science
here is a very famous philosopher in asian
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region in 10th century he is saying that he
learn the methods of calculation the indian
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methods of calculation from a vegetable vendor
so this was the day that the place value system
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really revolutionize calculation all over
the world this more
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modern quotation by laplace and gauss saying
that this is indeed one of the most wonderful
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discoveries in the history of mathematics
now by the time when you come to aryabhatiya
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in 500 ce it discusses the what is called
as what is called as parikarma logistics methods
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of calculation where square square root cube
cube root areas of triangle circle trapezium
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approximate value of pi computing sine tables
problems to do
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with interceptor arch in a circle progressions
rule of three arithmetic of fractions and
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finally something very interesting called
as kuttakara which was aryabhatta one invention
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is the method of solving linear indeterminate
equation so this is the kind of sine table
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that aryabhatta came up with and details later
on being systematically including india this
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is by govindaswamy in 9 century and this improved
table is due to
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madhava when we come to the later period we
have luminaries like to aryabhatta brahmagupta
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one of the most celebrated 25th in india in
varahamihira which is a compendium there is
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a chapter on human and there he is introducing
combinatorics idea and he is explaining that
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1820 various perfumes can be formed by choosing
4 out of a collection of 16 and to calculate
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the 16 c4 he gives a
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different kind of he is giving a different
kind of a table ahh here the first column
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natural integers the second column some of
natural integers the third column is the sum
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of sums of natural integers fourth column
is and its based upon the reconciliation which
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is equivalent to brahmasphutasiddhanta of
brahmagupta is a text on astronomy it has
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2 chapter in mathematics chapter 12 when 13
is called ganitagya and chapter 17 is called
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chapter 12 is ganitadhyaya 17 is ideas with
most ideas in algebra in brahmagupta for the
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first time find the arithmetic of negative
qualities calculations with zero and then
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detailed statement of equations and even introduction
of complicated equations known as the vargaprakrti
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which became a very important equation in
the indian mathematical tradition
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brahmagupta also gave very interesting result
such as this equation of the diagonals of
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a cyclic quadrilateral a quadrilateral which
is inscribed in a circle he gave a formula
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for the diagonals of a cyclic quadrilateral
in terms of the sites and an expression for
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the area of the circle quadrilateral which
is a generalization of the formula that perhaps
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all of you know as the heroines formula for
the area of a triangle brahmagupta
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course mentioned that this formula is applicable
to quadrilateral in triangle he gave some
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interesting properties of equations of the
kind these are called the varga prakyathi
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equation he was the first person to call me
later property called as bhavana the given
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1 solution you can go to another solution
we will discuss this later during the course
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but this
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enabled later on indian mathematician to work
out every systematic algorithm this one of
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the most famous algorithms in indian mathematics
called as cakravala and it enables you to
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solve equations this is a very famous problem
x square-61 y squared=1 you have to solve
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for x and y in integers and as you can see
this solutions are about 1p 1 7 p and 226
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million so these are very high numbers this
lowest solution of this equation after
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bhaskaracharya who is book 1150 solve this
equation by a very simple method this table
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tells you the method this problem again came
up 500 years later when bharma post this as
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a problem to the british mathematician ideas
of calculus started developing and they arose
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the context of astronomy the idea instantaneous
velocity become important because especially
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to understand the motion of moon one needed
to know ahh the rate of
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variation of its position and one found that
even the rate of change of its position was
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continuously changing and the idea of instantaneous
velocity arouse this way now there is a common
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misconception 6 years ago in modern times
that bhaskaracharya ii 11 ad was the last
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important mathematician in indian mathematics
afterwards the people were just repeating
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what was done in earlier books or they forgotten
mathematics all together it is only in the
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last
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56 years that works of later mathematicians
have been studied and understood and actually
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the picture is quite different ahh first of
all around 1200 works in mathematics started
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at caring in regional language ganitasarakaumudi
in prakrita vyavaharaganita in kannada pavuluriganitamu
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in telugu these are very important was written
around 13 century 12th century ganitakaumudi
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and narayana pandita is a great advance of
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bhaskaracharya lilavati a large part of course
will be devoted to study of that then there
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arouse a school in kerala which had been special
contributions to make the kerala school of
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astronomy initiated by madhava then parameswaran
then nilakantha somayaji they revise the older
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astronomical model and came up with a new
astronomical model but madhava is more well
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known for his discovery of infinite series
for pi sine and cosine and their the proofs
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of all
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hese results of madhava written down in a
very famous malayalam book called yuktibhasa
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written in 1530 mathematics continues in maharashtra
and kasi with scholars such as jnanaraja ganesa
198
00:22:43,880 --> 00:22:51,360
daivajna suryadasa they wrote proofs on bhaskara
result trigonometrically result were discovered
199
00:22:51,360 --> 00:22:59,280
by munisvara kamalakar savai jayasimha in
jaipur he built this 5 observatories which
200
00:22:59,280 --> 00:23:02,909
was very important at that time to correct
the older
201
00:23:02,909 --> 00:23:09,549
astronomical calculations the kerala school
also continued the last back was sankaravarman
202
00:23:09,549 --> 00:23:18,600
in 1830 there was an astronomer called candrasekhara
samanta in orissa who impact the game by traditional
203
00:23:18,600 --> 00:23:28,950
method all the major lunar inequalities in
1869 so just to tell you the kind of that
204
00:23:28,950 --> 00:23:34,400
narayana pandita even considered topics like
magic squares as serious mathematical topics
205
00:23:34,400 --> 00:23:38,730
and came up with very interesting way of constructing
magic
206
00:23:38,730 --> 00:23:44,240
squares several very new algorithms this is
what is called is the folding method of calculating
207
00:23:44,240 --> 00:23:50,230
magic squares this is the infinite series
for the ratio of the circumference to diameter
208
00:23:50,230 --> 00:23:56,610
discovered by by madhava the kerala mathematician
he not only discovered the infinite series
209
00:23:56,610 --> 00:24:04,860
that is a slowly convergent series that 1-1/3+1/5/1/7
if we calculate 50 tons of that series you
210
00:24:04,860 --> 00:24:11,870
get only one decimal place in the expansion
of pi so madhava at the same time gave what
211
00:24:11,870 --> 00:24:13,200
are known as the end correction
212
00:24:13,200 --> 00:24:19,120
terms so this is the first end connection
term due to madhava then there is another
213
00:24:19,120 --> 00:24:24,409
end correction term it is this end correction
terms which give you more accurate and more
214
00:24:24,409 --> 00:24:30,510
accurate result even if you sum only 50 terms
in the madhava series incidentally that series
215
00:24:30,510 --> 00:24:39,320
due to madhava is also known as the series
because it discovered by 1674 so using his
216
00:24:39,320 --> 00:24:41,110
connection madhava was
217
00:24:41,110 --> 00:24:47,110
able to give the value of pi correctly to
30 11 decimal places just by using 50 terms
218
00:24:47,110 --> 00:24:56,059
in his series with that end correction term
so we can briefly sketch this history of pi
219
00:24:56,059 --> 00:25:01,840
as typical of the way mathematics developed
across different cultures ahh please see aryabhatta's
220
00:25:01,840 --> 00:25:07,120
value 3 1416 which is activate up to 4 decimal
places that sulvasutra values which is activated
221
00:25:07,120 --> 00:25:12,820
up to 1 decimal places the jaina text use
root 10 archimedes give this
222
00:25:12,820 --> 00:25:19,821
standard is equality 3 10/71 less than pi
less than 3 1/7 the chinesh mathematician
223
00:25:19,821 --> 00:25:29,029
tsu chhung chih had this 355/113 which is
accurate up to nearly 7 6-7 decimal places
224
00:25:29,029 --> 00:25:35,980
but fact to this please see madhava coming
up 11 decimal places between aryabhatta to
225
00:25:35,980 --> 00:25:41,010
madhava then madhavas result was based upon
infinity series all these most of these results
226
00:25:41,010 --> 00:25:48,730
are actually based upon ahh root force calculation
with approximating the area of a circle by
227
00:25:48,730 --> 00:25:58,980
polygon al kasi etc newton again came up with
an infinite series around 1665 then the various
228
00:25:58,980 --> 00:26:04,510
other thing but we can see in recent time
ramanujan in 1914 came up with a very interesting
229
00:26:04,510 --> 00:26:12,770
series for pi using modular equation and that
created a small recorded that act at 1980s
230
00:26:12,770 --> 00:26:17,970
that people calculated pi to about 17 million
decimal places today's achievement is about
231
00:26:17,970 --> 00:26:22,560
5 trillion but equal important of this exact
results of
232
00:26:22,560 --> 00:26:28,480
pi you can see madhava all these exact result
which was later on repeated by others james
233
00:26:28,480 --> 00:26:36,230
gregory tan inverse series with series short
series all these are contained in madhavas
234
00:26:36,230 --> 00:26:44,480
paper this is the series given by ramanujan
in his 1914 paper the idea of instantaneous
235
00:26:44,480 --> 00:26:50,140
velocity also lead to ahh more complicated
derivative the derivative of sine function
236
00:26:50,140 --> 00:26:54,429
as a cosine function was well known by the
time of bhaskara nilakantha is
237
00:26:54,429 --> 00:27:00,190
formulating that derivative of sine inverse
function as 1/square root of 1-x square in
238
00:27:00,190 --> 00:27:07,429
this words now again till about 56 years ago
people had study only i mean the modern scholars
239
00:27:07,429 --> 00:27:13,659
had studies only the basic text of indian
mathematics so they had the sort of idea that
240
00:27:13,659 --> 00:27:18,280
indian somehow enhance lot of results but
they did not seem to have any method for arriving
241
00:27:18,280 --> 00:27:22,909
at this result or at least those messages
were very obscure so it
242
00:27:22,909 --> 00:27:27,740
is only in the last 56 years that many of
the common trees to the original text people
243
00:27:27,740 --> 00:27:34,580
started studying traditionally such issues
that how to obtain results how to understand
244
00:27:34,580 --> 00:27:40,600
them etc have been dealt with in detail bhaskara
commentary this is not just super mathematics
245
00:27:40,600 --> 00:27:46,360
that if you pick up any basic text even bhagavad
gita to understand ahh it in a very serious
246
00:27:46,360 --> 00:27:51,180
manner you have to take requests to the detail
common trees which are written on them and
247
00:27:51,180 --> 00:27:52,180
this
248
00:27:52,180 --> 00:27:55,799
commentaries continue to be returned till
recent times they played a very vital role
249
00:27:55,799 --> 00:28:01,180
in the traditional schema learning as per
mathematics is concerned it is in this comment
250
00:28:01,180 --> 00:28:07,470
that we find what are known as upapatti or
uptis they are something similar to demonstration
251
00:28:07,470 --> 00:28:13,450
a rational of proofs in mathematics you one
of the oldest words available words which
252
00:28:13,450 --> 00:28:21,029
has upapatii is a bhaskara 1 commentary on
aryabhatta but of course the most detailed
253
00:28:21,029 --> 00:28:30,890
exposition of upapati is found in the malayalam
text yuktibhasa written in 1530 now as you
254
00:28:30,890 --> 00:28:39,350
text upapatti what was the upapatti's suppose
to do what is the nature of this this was
255
00:28:39,350 --> 00:28:47,570
captured by this words of baskaran upapati
mean without the proof a mathematician will
256
00:28:47,570 --> 00:28:56,240
not be considered as a scholarly mathematician
in any assembly of mathematician any doubt
257
00:28:56,240 --> 00:29:01,870
regarding the result that he is enunciating
so for this reason that
258
00:29:01,870 --> 00:29:07,600
i am going to discuss upapatti's are true
that is what bhaskara is explaining in his
259
00:29:07,600 --> 00:29:15,419
commentary on siddhanta shiromani the same
point is repeated across ganesa is follower
260
00:29:15,419 --> 00:29:21,419
in the tradition of bhaskara is writing a
commentary on leelavati in 1540 explaining
261
00:29:21,419 --> 00:29:32,760
this proves again to that person who does
not know upapatti will not be without confusion
262
00:29:32,760 --> 00:29:39,020
normally be considered as a serious mathematician
now so the basic
263
00:29:39,020 --> 00:29:44,929
purpose of a upapatti is sort of clearly stated
to be to remove confusion and doubt regarding
264
00:29:44,929 --> 00:29:52,480
the validity and to obtain ascent ahh in the
community or something like sending a paper
265
00:29:52,480 --> 00:29:58,240
and getting it period and getting it publish
it does it mean that result is going to stand
266
00:29:58,240 --> 00:30:06,080
for all times for all ages ahh that was the
ideal of proof in the european tradition that
267
00:30:06,080 --> 00:30:11,430
does not seem to be the kind of ideal that
the indians are initiated by doing mathematics
268
00:30:11,430 --> 00:30:12,510
in fact
269
00:30:12,510 --> 00:30:19,530
the detail study of proofs in indian mathematics
shows that there are the differences between
270
00:30:19,530 --> 00:30:24,610
the idea of proof as we know from the greek
or european tradition and the idea of upapatti
271
00:30:24,610 --> 00:30:28,990
in indian mathematics first of all the indian
mathematician is a very clear that proofs
272
00:30:28,990 --> 00:30:35,140
are needed upapatti are needed ahh result
even if verified in 100s of cases does not
273
00:30:35,140 --> 00:30:39,660
mean that it is proved in mathematic so only
when you can give some logical argument or
274
00:30:39,660 --> 00:30:40,820
some other argument you
275
00:30:40,820 --> 00:30:46,160
can you say that it is a valid mathematical
result and several commentary are written
276
00:30:46,160 --> 00:30:52,570
ahh listing such upapattis when the upapattis
like as we know proofs in modern arithmetic
277
00:30:52,570 --> 00:30:57,640
ahh they are written in a sequence that to
go from known result to new result and from
278
00:30:57,640 --> 00:31:06,399
them to let other result so you will have
a sequence of establishing results and the
279
00:31:06,399 --> 00:31:12,019
understanding is that it is by giving proves
we are clear how the result is to be applied
280
00:31:12,019 --> 00:31:13,019
and
281
00:31:13,019 --> 00:31:17,940
understood the proofs may many times depend
upon experiment this something which is new
282
00:31:17,940 --> 00:31:23,330
ahh we may be doing it another mathematics
teaching but euclidean ideal of proof is that
283
00:31:23,330 --> 00:31:28,460
to prove something is very abstracted should
not be dependent on experimentation should
284
00:31:28,460 --> 00:31:32,950
not be depend on even our understanding what
is the nature of the mathematical object but
285
00:31:32,950 --> 00:31:38,360
the ahh indian proofs were always ahh they
could involve experimentation they could involve
286
00:31:38,360 --> 00:31:39,360
an
287
00:31:39,360 --> 00:31:43,230
understanding of the explicit use of the nature
of the object and another crucial things is
288
00:31:43,230 --> 00:31:50,059
that what is called the proof by contradiction
the which is called in indian mathematics
289
00:31:50,059 --> 00:31:57,929
that was employed the was employed ahh to
understand the non existence of certain mathematical
290
00:31:57,929 --> 00:32:04,630
quantities but it was not employed to established
the existence of a mathematical object whose
291
00:32:04,630 --> 00:32:08,840
existence would not otherwise be accessible
to us by any other
292
00:32:08,840 --> 00:32:17,070
means so (tl) non considered as a independent
from so existence of quantities cannot be
293
00:32:17,070 --> 00:32:22,929
established by nearly proofing that their
non existence is inconsistent with whatever
294
00:32:22,929 --> 00:32:29,770
we know but by giving a means as an access
to the way there existence can be understood
295
00:32:29,770 --> 00:32:37,950
by us which is something known as the constructive
philosophy which is mathematic and there is
296
00:32:37,950 --> 00:32:43,070
no ideal the proofs will give it that curable
demonstration or will give
297
00:32:43,070 --> 00:32:49,770
this the absolute truth of mathematical proposition
there was no idea that you fix one set of
298
00:32:49,770 --> 00:32:56,140
postulates once in for all in derive all the
results and by so many symbolism and symbolic
299
00:32:56,140 --> 00:33:01,700
techniques were used formalization of mathematics
was not something that is attempted in indian
300
00:33:01,700 --> 00:33:06,299
mathematic now coming to more contemporary
kinds this issue of proof we tell something
301
00:33:06,299 --> 00:33:10,890
very crucial in understanding the mathematic
of srinivasa ramanujan
302
00:33:10,890 --> 00:33:18,210
then ramanujan sent his result in 1930 in
a long method to high if it is 100 120 results
303
00:33:18,210 --> 00:33:23,470
hard immediately response by saying this all
kind looking very interesting but where are
304
00:33:23,470 --> 00:33:28,100
the proofs you please send me the proofs of
all these results ahh of course they was not
305
00:33:28,100 --> 00:33:32,149
so trivial that hardly could prove it for
himself the straight away on a piece of paper
306
00:33:32,149 --> 00:33:36,480
or something like that when did the proofs
be given and ramanujan there is a very famous
307
00:33:36,480 --> 00:33:39,159
let he send hardly in 1930
308
00:33:39,159 --> 00:33:43,919
saying that he has a systematic method for
deriving all the results but that cannot be
309
00:33:43,919 --> 00:33:49,540
explained in a short communication and he
thinks that he has a new methodology for doing
310
00:33:49,540 --> 00:33:54,720
things and he anywhere but he says that why
do not you just some of this results and can
311
00:33:54,720 --> 00:33:59,669
we check what i am writing his really and
that should convince you that there is something
312
00:33:59,669 --> 00:34:03,750
interesting in what i am doing now
313
00:34:03,750 --> 00:34:22,099
issues important because ahh finds where there
is this notebook of ramanujan ahh which is
314
00:34:22,099 --> 00:34:29,100
ahh set of all results that he noted hire
to going to england and later analysis in
315
00:34:29,100 --> 00:34:36,379
the last 2530 shows that there are more than
3000 results this notebooks contain are the
316
00:34:36,379 --> 00:34:41,230
initially thought that two thirds of them
where already well known but now the understanding
317
00:34:41,230 --> 00:34:45,293
is more than two thirds was not known it the
time ramanujan was recording this results
318
00:34:45,293 --> 00:34:47,859
in the notebook
319
00:34:47,859 --> 00:34:53,950
and almost all the results are correct and
there is no more than 5 to 10 or incorrect
320
00:34:53,950 --> 00:34:59,079
this is the current assessment of the results
that ramanujan wrote down in his notebook
321
00:34:59,079 --> 00:35:04,410
there is of course a notebook of the work
that he was doing in the last year of his
322
00:35:04,410 --> 00:35:11,859
life 1919 to 20 ahh which was lost seemingly
and it was recorded in 1975 in the trinity
323
00:35:11,859 --> 00:35:18,210
college library by mr g e andrews it is called
the lost notebook and result in that are still
324
00:35:18,210 --> 00:35:19,210
being
325
00:35:19,210 --> 00:35:23,650
established by the mathematician of present
day and this contains full lot of results
326
00:35:23,650 --> 00:35:30,710
like this so what i was trying to say was
that greco-european tradition of mathematics
327
00:35:30,710 --> 00:35:38,720
almost equate mathematics with proof and the
way mathematical results of discovery therefore
328
00:35:38,720 --> 00:35:46,690
is hardly understood which may be termed as
intuition natural genius etc and there is
329
00:35:46,690 --> 00:35:50,869
an understanding that mathematical results
of non empirical and therefore there is
330
00:35:50,869 --> 00:35:58,700
no access to them except by logical argumentation
of course there are philosopher of mathematics
331
00:35:58,700 --> 00:36:05,010
to do argue that this philosophy of mathematics
is confusing the barrel ahh this philosophy
332
00:36:05,010 --> 00:36:10,560
does not explain most of history of mathematics
today mathematics is done either it was done
333
00:36:10,560 --> 00:36:17,089
in earlier life or even mathematics is being
done present in indian tradition the understanding
334
00:36:17,089 --> 00:36:23,180
was that proof is only one of the aspects
of mathematics important ahh
335
00:36:23,180 --> 00:36:27,290
mathematical result does not part of to be
nonempirical mathematics was not thought out
336
00:36:27,290 --> 00:36:34,410
to be a science different from other sciences
it results were equally contestable and falsifiable
337
00:36:34,410 --> 00:36:39,200
and they could be validated in diverse days
the proof was important but they were more
338
00:36:39,200 --> 00:36:47,730
for obtaining as an for once result ahh so
the process of mathematical discovery in the
339
00:36:47,730 --> 00:36:51,500
mathematical justification or in some unicell
in where the indian have
340
00:36:51,500 --> 00:36:58,470
understood mathematics long time ago when
ramanujan letter arrived in england the conclusion
341
00:36:58,470 --> 00:37:08,010
that it and it would had but one of his friends
that have discovered second newton in a hindu
342
00:37:08,010 --> 00:37:15,890
but if some comparison is to be made regarding
ramanujan ahh he is more in the line up madhava
343
00:37:15,890 --> 00:37:21,700
both in the kind of topics like infinity transformations
of them and continued fractions and transformations
344
00:37:21,700 --> 00:37:26,880
of them ahh handling iteration and indeed
success of
345
00:37:26,880 --> 00:37:33,550
the great genius madhava who was one of the
pioneer of calculation i tried to extract
346
00:37:33,550 --> 00:37:42,750
review of recent book on mathematics in india
by david mumford the well known the main point
347
00:37:42,750 --> 00:37:48,560
was emphasize that by studying indian mathematics
of the history of mathematics in india what
348
00:37:48,560 --> 00:37:55,770
one can understand is that indian mathematics
can be done in different ways the views of
349
00:37:55,770 --> 00:37:59,190
mathematics can be boot in many different
ways and have secret
350
00:37:59,190 --> 00:38:04,760
is out of that and so they were in india and
one should not just confused the fact that
351
00:38:04,760 --> 00:38:09,820
absence of rigorous mathematics in the greek
style means that the rest is not mathematics
352
00:38:09,820 --> 00:38:16,980
at all when he caution that most of interesting
mathematics ahh that we used today which was
353
00:38:16,980 --> 00:38:24,090
developed in 16 17 18 century was indeed done
by abandon the greek term of doing mathematics
354
00:38:24,090 --> 00:38:29,000
ahh this is the kind of understanding that
scholars are arriving at the importance of
355
00:38:29,000 --> 00:38:30,000
knowing a
356
00:38:30,000 --> 00:38:36,510
different tradition of mathematics like the
indian tradition another interesting in this
357
00:38:36,510 --> 00:38:44,050
the question of ahh the history of science
in recent times that ever seen the work of
358
00:38:44,050 --> 00:38:51,630
needham it has generally been understood that
till 16 century ahh the chinese science and
359
00:38:51,630 --> 00:38:57,560
technology seem to be considerable advanced
over science and technology in europe and
360
00:38:57,560 --> 00:39:04,260
then needham showed the question that needham
almost made it an important focus was why
361
00:39:04,260 --> 00:39:10,119
modern science did not emerge in china and
did not emerge in non-western societies now
362
00:39:10,119 --> 00:39:17,369
when we study mathematics in india for instance
notice that many of hallmarks of modern science
363
00:39:17,369 --> 00:39:24,820
such as development of calculus infinite series
etc are development of you astronomical ahh
364
00:39:24,820 --> 00:39:31,960
models of the planet which system ahh they
were all there in kerala that of 14 15 16
365
00:39:31,960 --> 00:39:37,940
century so a very crucial question that we
should understand
366
00:39:37,940 --> 00:39:43,680
is why science did not flourish in non-western
societies that is 16th century and it is even
367
00:39:43,680 --> 00:39:49,369
more important but today's purpose ahh to
have some idea how science would have developed
368
00:39:49,369 --> 00:39:54,790
ahh how the science today would have been
if the non-western societies had continued
369
00:39:54,790 --> 00:39:59,680
developing science along the lines that they
have laying down for themselves earlier ahh
370
00:39:59,680 --> 00:40:04,180
maybe with many modifications maybe with some
transformations in interaction
371
00:40:04,180 --> 00:40:11,579
with modern science developed in europe and
subsequent time it is only by that kind of
372
00:40:11,579 --> 00:40:17,000
speculation we can come to some understanding
on of the great genius of modern times in
373
00:40:17,000 --> 00:40:25,230
india such as ramanujan bose prafulla chandra
roy raman and many others so to summarise
374
00:40:25,230 --> 00:40:36,240
the development of mathematics in india that
the main thing was that the complex mathematical
375
00:40:36,240 --> 00:40:40,180
problems were not send even if complete solutions
to them were not
376
00:40:40,180 --> 00:40:46,650
found approximate ahh less than perfect solutions
were accepted and then developed into better
377
00:40:46,650 --> 00:40:55,690
and better solution and the idea was always
was in simplicity of mathematical procedure
378
00:40:55,690 --> 00:41:01,230
and by this indians were able to do quite
a bit they could get the basic () geometry
379
00:41:01,230 --> 00:41:07,089
by the time of sulvasutras they could establish
most of our arithmetic algebra geometry and
380
00:41:07,089 --> 00:41:12,790
trigonometry by the time of aryabhatta by
the time of bhaskara ii they could
381
00:41:12,790 --> 00:41:20,410
solve complicated quadratic indeterminate
equations or by 14 15 century calculus exact
382
00:41:20,410 --> 00:41:29,260
series were sine and very accurate sine tables
which all very important perform so the crucial
383
00:41:29,260 --> 00:41:36,920
thing is explicitly algorithmic and computational
nature of indian mathematics and this seems
384
00:41:36,920 --> 00:41:42,260
to have persisted that till recent times and
to some extent of srinivasa ramanujan as i
385
00:41:42,260 --> 00:41:53,900
told you could be thought of as a traditional
indian methodology and perhaps is important
386
00:41:53,900 --> 00:41:54,900
that we
387
00:41:54,900 --> 00:42:00,500
should have a detailed understanding of the
development of mathematics in india ahh to
388
00:42:00,500 --> 00:42:06,320
understand the way indians approached many
complex problems even in other sine and we
389
00:42:06,320 --> 00:42:12,589
let to say ahh it very important that we should
teach the highlights of this great tradition
390
00:42:12,589 --> 00:42:18,370
of mathematics to our students in schools
and colleges and i think courses like this
391
00:42:18,370 --> 00:42:28,530
will help in sort of formulating that kind
of a work so with that i i complete this initial
392
00:42:28,530 --> 00:42:30,059
overview and thank you very much