good morning this is the first lecture of
this course ahh which is being given on mathematics in india from vedic period to modern times
it is a novel course which tries to trace the way mathematics developed in india ahh
the first talk is an overview talk in this i will try to highlight those periods is there
was a significant development of mathematics in india i will also try to summarise ahh
the special nature of athletics as a developed in india i would like to emphasize the algorithmic
way in which most problems in mathematics was considered in the indian tradition so
i am flashing the outline not going to read it out ahh we can just see the kind of topics
we are going to follow up ahh we will cover the development of indian mathematics in the
ancient period indicate some highlights during that period then the early classical period
say 500 bce to 500 ce which culminated in the birth of aryabhatta then the development mathematics
in the latest classical period from 500 in the common era to 1250 we will then go to
some uses of the highlights of what happened during the mediaeval period till about 1850
towards then we will discuss something about the nature of mathematics in india how mathematics
how was results like that and then about the contemporary period we will speak a little
bit about the srinivasa ramanujan finaly know that history so some
find out pervasive mathematics in india is this following statement from the ganita sara
sangraha of mahaviracharya if along six packs statement again when mahavirachary it got
by saying that mathematics is important in all areas when finally concludes by saying
to that is not provided by mathematics so that is kind of statement that mahaviracharya
is beating weather is it in astronomy or beat in architecture or beat in conjunction of
granite position kind course of moon logic quite the
grammar so it says all purpose statement of mahaviracharya that mathematics provides all
aspects all subject this quotation from the 9 century world called ganita sara sangraha
or mahaviracarya so ganita stands for calculation competition to is a statement due to ganesa
daivajna is a commentator of lilavati and therefore we can expect that indian mathematical text really abound in
rules to describe systematic and efficient procedure for calculation just to give you
an example we will go to a very ancient rule this is given by bhaskara i this is in this
commentary to aryabhatiya it is just a rule for calculating the square of a number to
so we can see the kind of calculation was talking about to take any number 125 first
you square the last number multiply the other numbers by 2 and the last number so
you get this row then move away remove one digit square the next number multiply by 2
and that number the next row and finally square the last number remove one number add all
of them the important thing to realises even this very ancient rule written in 1619 ad
actually uses n*n-1/2 multiplication to calculate the square of a number a n digit number multiplied
by another n digit number we will n square multiplication but since we are squaring the
same number is 2ab like that comes in and so you are having an optimal algorithm for
square this is the indian mathematicians always right to give the best possible way of the
best possible procedure for doing a calculation now the algorithm itself as a history ahh
it was the name given to the indian methods of doing calculation when we coordinates when
the name of central asia mathematician call who in the ninth century wrote a book on indian
methods of calculation that is methods of calculation using the decimal place value
system and that book was called algorithm latin version of that book is available the
original arabic version is not available and this was the book that introduced the decimal
place value notation to the arabic world and later on to the european world and so the
people who followed this was calculation were called algorithm and the algorithm comes from
() and this algorithm factor is not something very specific to mathematics impact it provides
all indian sciences most of indian disciplines sastras as we call them they do not present
a series of propositions they normally give you a set of rules a set of procedure which
tell you how to systematically accomplish something so the rules given in sastras are
usually called as vidhi kriya prakriya sadhana parikarma karana these are the names
and these rules are what are usually formulated as sutras so disciplines the ethical disciplines
in india they provide systematic rules of procedure rather than a set of propositions
and the for the most canonical such systematic text in india is the great grammar written
by panini called astadhyayi in fact most other disciplines and especially mathematics is
extensively influenced by the method of panini please use symbolic and technical devices
recursive and generative formalism and this system of convention that govern rule application
and rule interaction all these go back to panini and it has deep influence you are not
the modern discipline of linked list in fact many scholar actually acknowledge that place
panini holds in indian tradition plays taht something analogous to the place you could
hold in the ecuclidean tradition and here is a quotation from stall
where he saying panini is also dividing systematically sanskrit occurrences from a set of rules and
euclid is also deriving a set of propositions from a collection of axioms but the world
deriving will have 2 different mean means in this 2 context panini is actually generating
valid occurrences of sanskrit is not proving theorems euclid is demonstration is proving
theorems in mathematics from a set of postulates so in ancient period the ugliest
text in mathematics available are the text on construction of higher alters the vedis
these are the sulvasutras these are the oldest texts of geometry in teh world they give procedures
for construction in transformation of geometrical figures then there are ancient astronomical
siddhantas which deal with astronomy when we come to the classical period starting from
panini we then have the chandahsutra of pingala which initiated combinatorics we have
some mathematics in the jaina tradition in the jaina then more crucially the idea of
0 and decimal place value system developed in this period and all these terminated in
the mathematics and astronomy that is found in the text aryabhatiya aryabhatta which was
written in 499 of the common error most of the standard procedure in arithmetic algebra
geometry trigonometry were perfected and many more things which was used in astronomy
like the indeterminate equations sign tables all these things were perfected by the time
of aryabhatta so the ancient sulvasutras deal with lot of things units of measurement marking
directions construction of rectangle square trapezium transformation of square and it
has the first oldest statement of geometry the theorem which we attribute phithogorous
commonly it is called the bhuja-koti-karna-nyaya in later indian mathematical text it is the sum
of the two sides of a rectangle the square sum of the squares of two sides of a rectangle
is equal to the square of the diagonal this is the rule i stated in baudhayana sulvasutras
to there are even more complicated rules of adding squares then there is a rule for approximate
conversion of square into a circle which leads to a value of 5 around 3 08 then there is
a very interesting formula for square root of 2 is
called dvikarana in it is accurate of 2 several decimal places that you can see finally all
this geometry is used in constructing all types this is the rule in katyana sulvasutra
the problem is how to construct a square which is n times ahh the area of a given square
and katyana sulvasutra gives a very interesting geometrical formula n+1a/2 whole squared-n-1a/2
whole square is na square it is using this very interesting algebra it result
to calculate the side of a square which is n times in area of the given square pingala
sutra are the combinatorics and these are a very interesting diagram known as the meru
prastara which appears in pinglas it gives you the what we now call as the binomial coefficients
ncr they arise very naturally when you want to count how many metres are there which are
n syllables but in which are number of groups appears that is ncr as we shall see
later the decimal place value system arose in the ancient period the main thing about
the decimal place value system is that is an essential in algebraic concept the number
52038 written as 5 times 10 cube 2 times 10 square and 0 times 10+3 is something i think
to a algebraic polynomial 5x cube 2z+square+ 0x+3 it is this algebraic future of place
value system that enabled the indian mathematicians to give systematic and very interesting procedure
for making calculations and they became the standard methods of calculation all the world
over sometimes the indian books do give some special techniques also which are essentially
originating out of the place value systems for instance the forming the buddhivilasini
of ganesa daivajna lilavati it discusses what is currently popularly known as the vajrabhyasa
method of multiplication vertical and cross wise method multiplication the history of
decimal place value system goes back to the vedas they use the system to the base 10 very
naturally the upanisads talk of zero and infinity panini's astadhyayi has a notion of ahh lopa
which is i think what is called as this idea of in bauddha philosophy the idea of abhava
in the naiyayika philosophy pingala's chandahsustra uses a 0 as a marker which not a clear whether
at that time the idea of 0 as a number was no now soon enough the idea
of place value system became so common that philosophical works ahh such as vasumitras
with this text and even and yogasutra started explaining the speciality of the place value
system there is a quotation from the vyasabhasya on yogasutra to just as nad is understood
as a mother daughter-in-law or a sister ahh li which appears at different places that
is number 1 which appears at different places will have different
values and they got 10 so like this this issue became well known in the circles of philosophy
also and got discussed and one of the oldest place value system explicitly is in a book
called vrddhayanajataka written by sphujidhvaja around 270 in the common era and by the time
of aryabhatta aryabhatiya all calculations are formulated with the place value system
her an inscription in gwalior which is giving the number 270 it means it is appearing
here and the many other inscriptions in southeast asia in gwalior in various other places around
early 7 century ahh which give numbers in the place value system with 0 also now this
indian place value system acclaimed universally this statement in the 7th century by a syrians
who is this out of saying that the greek seem to think too much of themselves but they really
do not know the basic methods of calculation that indians
have discovered and they better know that the others also know something of science
here is a very famous philosopher in asian region in 10th century he is saying that he
learn the methods of calculation the indian methods of calculation from a vegetable vendor
so this was the day that the place value system really revolutionize calculation all over
the world this more modern quotation by laplace and gauss saying
that this is indeed one of the most wonderful discoveries in the history of mathematics
now by the time when you come to aryabhatiya in 500 ce it discusses the what is called
as what is called as parikarma logistics methods of calculation where square square root cube
cube root areas of triangle circle trapezium approximate value of pi computing sine tables
problems to do with interceptor arch in a circle progressions
rule of three arithmetic of fractions and finally something very interesting called
as kuttakara which was aryabhatta one invention is the method of solving linear indeterminate
equation so this is the kind of sine table that aryabhatta came up with and details later
on being systematically including india this is by govindaswamy in 9 century and this improved
table is due to madhava when we come to the later period we
have luminaries like to aryabhatta brahmagupta one of the most celebrated 25th in india in
varahamihira which is a compendium there is a chapter on human and there he is introducing
combinatorics idea and he is explaining that 1820 various perfumes can be formed by choosing
4 out of a collection of 16 and to calculate the 16 c4 he gives a different kind of he is giving a different
kind of a table ahh here the first column natural integers the second column some of
natural integers the third column is the sum of sums of natural integers fourth column
is and its based upon the reconciliation which is equivalent to brahmasphutasiddhanta of
brahmagupta is a text on astronomy it has 2 chapter in mathematics chapter 12 when 13
is called ganitagya and chapter 17 is called chapter 12 is ganitadhyaya 17 is ideas with
most ideas in algebra in brahmagupta for the first time find the arithmetic of negative
qualities calculations with zero and then detailed statement of equations and even introduction
of complicated equations known as the vargaprakrti which became a very important equation in
the indian mathematical tradition brahmagupta also gave very interesting result
such as this equation of the diagonals of a cyclic quadrilateral a quadrilateral which
is inscribed in a circle he gave a formula for the diagonals of a cyclic quadrilateral
in terms of the sites and an expression for the area of the circle quadrilateral which
is a generalization of the formula that perhaps all of you know as the heroines formula for
the area of a triangle brahmagupta course mentioned that this formula is applicable
to quadrilateral in triangle he gave some interesting properties of equations of the
kind these are called the varga prakyathi equation he was the first person to call me
later property called as bhavana the given 1 solution you can go to another solution
we will discuss this later during the course but this enabled later on indian mathematician to work
out every systematic algorithm this one of the most famous algorithms in indian mathematics
called as cakravala and it enables you to solve equations this is a very famous problem
x square-61 y squared=1 you have to solve for x and y in integers and as you can see
this solutions are about 1p 1 7 p and 226 million so these are very high numbers this
lowest solution of this equation after bhaskaracharya who is book 1150 solve this
equation by a very simple method this table tells you the method this problem again came
up 500 years later when bharma post this as a problem to the british mathematician ideas
of calculus started developing and they arose the context of astronomy the idea instantaneous
velocity become important because especially to understand the motion of moon one needed
to know ahh the rate of variation of its position and one found that
even the rate of change of its position was continuously changing and the idea of instantaneous
velocity arouse this way now there is a common misconception 6 years ago in modern times
that bhaskaracharya ii 11 ad was the last important mathematician in indian mathematics
afterwards the people were just repeating what was done in earlier books or they forgotten
mathematics all together it is only in the last 56 years that works of later mathematicians
have been studied and understood and actually the picture is quite different ahh first of
all around 1200 works in mathematics started at caring in regional language ganitasarakaumudi
in prakrita vyavaharaganita in kannada pavuluriganitamu in telugu these are very important was written
around 13 century 12th century ganitakaumudi and narayana pandita is a great advance of bhaskaracharya lilavati a large part of course
will be devoted to study of that then there arouse a school in kerala which had been special
contributions to make the kerala school of astronomy initiated by madhava then parameswaran
then nilakantha somayaji they revise the older astronomical model and came up with a new
astronomical model but madhava is more well known for his discovery of infinite series
for pi sine and cosine and their the proofs of all hese results of madhava written down in a
very famous malayalam book called yuktibhasa written in 1530 mathematics continues in maharashtra
and kasi with scholars such as jnanaraja ganesa daivajna suryadasa they wrote proofs on bhaskara
result trigonometrically result were discovered by munisvara kamalakar savai jayasimha in
jaipur he built this 5 observatories which was very important at that time to correct
the older astronomical calculations the kerala school
also continued the last back was sankaravarman in 1830 there was an astronomer called candrasekhara
samanta in orissa who impact the game by traditional method all the major lunar inequalities in
1869 so just to tell you the kind of that narayana pandita even considered topics like
magic squares as serious mathematical topics and came up with very interesting way of constructing
magic squares several very new algorithms this is
what is called is the folding method of calculating magic squares this is the infinite series
for the ratio of the circumference to diameter discovered by by madhava the kerala mathematician
he not only discovered the infinite series that is a slowly convergent series that 1-1/3+1/5/1/7
if we calculate 50 tons of that series you get only one decimal place in the expansion
of pi so madhava at the same time gave what are known as the end correction terms so this is the first end connection
term due to madhava then there is another end correction term it is this end correction
terms which give you more accurate and more accurate result even if you sum only 50 terms
in the madhava series incidentally that series due to madhava is also known as the series
because it discovered by 1674 so using his connection madhava was able to give the value of pi correctly to
30 11 decimal places just by using 50 terms in his series with that end correction term
so we can briefly sketch this history of pi as typical of the way mathematics developed
across different cultures ahh please see aryabhatta's value 3 1416 which is activate up to 4 decimal
places that sulvasutra values which is activated up to 1 decimal places the jaina text use
root 10 archimedes give this standard is equality 3 10/71 less than pi
less than 3 1/7 the chinesh mathematician tsu chhung chih had this 355/113 which is
accurate up to nearly 7 6-7 decimal places but fact to this please see madhava coming
up 11 decimal places between aryabhatta to madhava then madhavas result was based upon
infinity series all these most of these results are actually based upon ahh root force calculation
with approximating the area of a circle by polygon al kasi etc newton again came up with
an infinite series around 1665 then the various other thing but we can see in recent time
ramanujan in 1914 came up with a very interesting series for pi using modular equation and that
created a small recorded that act at 1980s that people calculated pi to about 17 million
decimal places today's achievement is about 5 trillion but equal important of this exact
results of pi you can see madhava all these exact result
which was later on repeated by others james gregory tan inverse series with series short
series all these are contained in madhavas paper this is the series given by ramanujan
in his 1914 paper the idea of instantaneous velocity also lead to ahh more complicated
derivative the derivative of sine function as a cosine function was well known by the
time of bhaskara nilakantha is formulating that derivative of sine inverse
function as 1/square root of 1-x square in this words now again till about 56 years ago
people had study only i mean the modern scholars had studies only the basic text of indian
mathematics so they had the sort of idea that indian somehow enhance lot of results but
they did not seem to have any method for arriving at this result or at least those messages
were very obscure so it is only in the last 56 years that many of
the common trees to the original text people started studying traditionally such issues
that how to obtain results how to understand them etc have been dealt with in detail bhaskara
commentary this is not just super mathematics that if you pick up any basic text even bhagavad
gita to understand ahh it in a very serious manner you have to take requests to the detail
common trees which are written on them and this commentaries continue to be returned till
recent times they played a very vital role in the traditional schema learning as per
mathematics is concerned it is in this comment that we find what are known as upapatti or
uptis they are something similar to demonstration a rational of proofs in mathematics you one
of the oldest words available words which has upapatii is a bhaskara 1 commentary on
aryabhatta but of course the most detailed exposition of upapati is found in the malayalam
text yuktibhasa written in 1530 now as you text upapatti what was the upapatti's suppose
to do what is the nature of this this was captured by this words of baskaran upapati
mean without the proof a mathematician will not be considered as a scholarly mathematician
in any assembly of mathematician any doubt regarding the result that he is enunciating
so for this reason that i am going to discuss upapatti's are true
that is what bhaskara is explaining in his commentary on siddhanta shiromani the same
point is repeated across ganesa is follower in the tradition of bhaskara is writing a
commentary on leelavati in 1540 explaining this proves again to that person who does
not know upapatti will not be without confusion normally be considered as a serious mathematician
now so the basic purpose of a upapatti is sort of clearly stated
to be to remove confusion and doubt regarding the validity and to obtain ascent ahh in the
community or something like sending a paper and getting it period and getting it publish
it does it mean that result is going to stand for all times for all ages ahh that was the
ideal of proof in the european tradition that does not seem to be the kind of ideal that
the indians are initiated by doing mathematics in fact the detail study of proofs in indian mathematics
shows that there are the differences between the idea of proof as we know from the greek
or european tradition and the idea of upapatti in indian mathematics first of all the indian
mathematician is a very clear that proofs are needed upapatti are needed ahh result
even if verified in 100s of cases does not mean that it is proved in mathematic so only
when you can give some logical argument or some other argument you can you say that it is a valid mathematical
result and several commentary are written ahh listing such upapattis when the upapattis
like as we know proofs in modern arithmetic ahh they are written in a sequence that to
go from known result to new result and from them to let other result so you will have
a sequence of establishing results and the understanding is that it is by giving proves
we are clear how the result is to be applied and understood the proofs may many times depend
upon experiment this something which is new ahh we may be doing it another mathematics
teaching but euclidean ideal of proof is that to prove something is very abstracted should
not be dependent on experimentation should not be depend on even our understanding what
is the nature of the mathematical object but the ahh indian proofs were always ahh they
could involve experimentation they could involve an understanding of the explicit use of the nature
of the object and another crucial things is that what is called the proof by contradiction
the which is called in indian mathematics that was employed the was employed ahh to
understand the non existence of certain mathematical quantities but it was not employed to established
the existence of a mathematical object whose existence would not otherwise be accessible
to us by any other means so (tl) non considered as a independent
from so existence of quantities cannot be established by nearly proofing that their
non existence is inconsistent with whatever we know but by giving a means as an access
to the way there existence can be understood by us which is something known as the constructive
philosophy which is mathematic and there is no ideal the proofs will give it that curable
demonstration or will give this the absolute truth of mathematical proposition
there was no idea that you fix one set of postulates once in for all in derive all the
results and by so many symbolism and symbolic techniques were used formalization of mathematics
was not something that is attempted in indian mathematic now coming to more contemporary
kinds this issue of proof we tell something very crucial in understanding the mathematic
of srinivasa ramanujan then ramanujan sent his result in 1930 in
a long method to high if it is 100 120 results hard immediately response by saying this all
kind looking very interesting but where are the proofs you please send me the proofs of
all these results ahh of course they was not so trivial that hardly could prove it for
himself the straight away on a piece of paper or something like that when did the proofs
be given and ramanujan there is a very famous let he send hardly in 1930 saying that he has a systematic method for
deriving all the results but that cannot be explained in a short communication and he
thinks that he has a new methodology for doing things and he anywhere but he says that why
do not you just some of this results and can we check what i am writing his really and
that should convince you that there is something interesting in what i am doing now issues important because ahh finds where there
is this notebook of ramanujan ahh which is ahh set of all results that he noted hire
to going to england and later analysis in the last 2530 shows that there are more than
3000 results this notebooks contain are the initially thought that two thirds of them
where already well known but now the understanding is more than two thirds was not known it the
time ramanujan was recording this results in the notebook and almost all the results are correct and
there is no more than 5 to 10 or incorrect this is the current assessment of the results
that ramanujan wrote down in his notebook there is of course a notebook of the work
that he was doing in the last year of his life 1919 to 20 ahh which was lost seemingly
and it was recorded in 1975 in the trinity college library by mr g e andrews it is called
the lost notebook and result in that are still being established by the mathematician of present
day and this contains full lot of results like this so what i was trying to say was
that greco-european tradition of mathematics almost equate mathematics with proof and the
way mathematical results of discovery therefore is hardly understood which may be termed as
intuition natural genius etc and there is an understanding that mathematical results
of non empirical and therefore there is no access to them except by logical argumentation
of course there are philosopher of mathematics to do argue that this philosophy of mathematics
is confusing the barrel ahh this philosophy does not explain most of history of mathematics
today mathematics is done either it was done in earlier life or even mathematics is being
done present in indian tradition the understanding was that proof is only one of the aspects
of mathematics important ahh mathematical result does not part of to be
nonempirical mathematics was not thought out to be a science different from other sciences
it results were equally contestable and falsifiable and they could be validated in diverse days
the proof was important but they were more for obtaining as an for once result ahh so
the process of mathematical discovery in the mathematical justification or in some unicell
in where the indian have understood mathematics long time ago when
ramanujan letter arrived in england the conclusion that it and it would had but one of his friends
that have discovered second newton in a hindu but if some comparison is to be made regarding
ramanujan ahh he is more in the line up madhava both in the kind of topics like infinity transformations
of them and continued fractions and transformations of them ahh handling iteration and indeed
success of the great genius madhava who was one of the
pioneer of calculation i tried to extract review of recent book on mathematics in india
by david mumford the well known the main point was emphasize that by studying indian mathematics
of the history of mathematics in india what one can understand is that indian mathematics
can be done in different ways the views of mathematics can be boot in many different
ways and have secret is out of that and so they were in india and
one should not just confused the fact that absence of rigorous mathematics in the greek
style means that the rest is not mathematics at all when he caution that most of interesting
mathematics ahh that we used today which was developed in 16 17 18 century was indeed done
by abandon the greek term of doing mathematics ahh this is the kind of understanding that
scholars are arriving at the importance of knowing a different tradition of mathematics like the
indian tradition another interesting in this the question of ahh the history of science
in recent times that ever seen the work of needham it has generally been understood that
till 16 century ahh the chinese science and technology seem to be considerable advanced
over science and technology in europe and then needham showed the question that needham
almost made it an important focus was why modern science did not emerge in china and
did not emerge in non-western societies now when we study mathematics in india for instance
notice that many of hallmarks of modern science such as development of calculus infinite series
etc are development of you astronomical ahh models of the planet which system ahh they
were all there in kerala that of 14 15 16 century so a very crucial question that we
should understand is why science did not flourish in non-western
societies that is 16th century and it is even more important but today's purpose ahh to
have some idea how science would have developed ahh how the science today would have been
if the non-western societies had continued developing science along the lines that they
have laying down for themselves earlier ahh maybe with many modifications maybe with some
transformations in interaction with modern science developed in europe and
subsequent time it is only by that kind of speculation we can come to some understanding
on of the great genius of modern times in india such as ramanujan bose prafulla chandra
roy raman and many others so to summarise the development of mathematics in india that
the main thing was that the complex mathematical problems were not send even if complete solutions
to them were not found approximate ahh less than perfect solutions
were accepted and then developed into better and better solution and the idea was always
was in simplicity of mathematical procedure and by this indians were able to do quite
a bit they could get the basic () geometry by the time of sulvasutras they could establish
most of our arithmetic algebra geometry and trigonometry by the time of aryabhatta by
the time of bhaskara ii they could solve complicated quadratic indeterminate
equations or by 14 15 century calculus exact series were sine and very accurate sine tables
which all very important perform so the crucial thing is explicitly algorithmic and computational
nature of indian mathematics and this seems to have persisted that till recent times and
to some extent of srinivasa ramanujan as i told you could be thought of as a traditional
indian methodology and perhaps is important that we should have a detailed understanding of the
development of mathematics in india ahh to understand the way indians approached many
complex problems even in other sine and we let to say ahh it very important that we should
teach the highlights of this great tradition of mathematics to our students in schools
and colleges and i think courses like this will help in sort of formulating that kind
of a work so with that i i complete this initial overview and thank you very much