Since ancient times, man has felt the need to count objects around him: for this he invented the numbers. Soon, the need to count has been joined to perform operations on numbers. Thus was born the arithmetic, the basic part of mathematics that deals with the study of numbers and operations that can be performed on them. The development of arithmetic has followed several milestones, which have led to the introduction of new numbers in addition to the natural numbers used to count: integer relative, rational numbers, real numbers

The term derives from the Greek word arithmetikè arithmetic, itself formed from the terms arithmos ("number") and tèchne ("technical"), and means "art of numbers." The arithmetic is, in fact, the art of combining the numbers according to different procedures, calls operations. It deals with the basic operations: addition (+), subtraction (-), multiplication (×) and division (:), fractions, extractions and root of logarithms.

The first to use the word 'arithmetic' were the disciples of Pythagoras. The Pythagoreans have established the distinction between even numbers and odd numbers and discovered the primes, ie those divisible only for the unit and for themselves. Around 300 BC Euclid, has exposed these and other concepts of arithmetic and geometry in his most famous work, the Elements. In later centuries have been interested arithmetic Archimedes and Eratosthenes, the inventor of the sieve, a method for finding prime numbers smaller than an entire assigned.

count

They are called natural numbers positive integers, ie 1, 2, 3, 4, ... The natural numbers were the first to be invented, driven by the need to count objects, and the act of counting, that is, to indicate the number of objects belonging to a certain set, started to mathematics.

In arithmetic, an operation is a process which combines two elements a third element, the result of the operation. The addition, the fundamental operation of arithmetic, is, in practice, the continuation of the act of counting. The continuity between the act of counting and arithmetic operations is reflected also in natural language, in which the expression arithmetic indicates precisely the execution of arithmetic operations.

Dear Sundarapandian Vaidyanathan,

first of all thanks a lot! Your question is very interesting and very well argued. Often Indian mathematical history is not so known, and all the books authors refers on occidental history. This is a pity, because your history, or Chinese math history is one of the most important in the world. So thanks a lot for your suggestions, I'll read better your reference, and I'll propose that readings to my classroom.

Best regard,

Pierantozzi Mariano.

Two British researchers challenged the conventional history of mathematics in June when they reported having evidence that the infinite series, one of the core concepts of calculus, was first developed by Indian mathematicians in the 14th century. They also believe they can show how the advancement may have been passed along to Isaac Newton and Gottfried Wilhelm Leibniz, who are credited with independently developing the concept some 250 years later.

http://discovermagazine.com/2008/jan/calculus-was-developed-in-medieval-india

After posting the query to friends in Research-Gate, it occurred to me that mathematical rigour was very much missing in my general query and excerpts, and it made only casual reading.

After seeing Krishnan's excellent link, which is also somewhat meant for a general article only, I have decided to give a technical link, and interested fellow scientists can take a glimpse at the interesting problems developed 1000 years ago.. According to this link, Manjula (932) was the first Hindu astronomer to state that the difference of the sines as

         sin w' - sin w = (w' - w) cos w,

where (w' - w) is small. Bhaskara II applied this formula in his astronomical results, and developed some of his own results (prototype result of now what is known as "mean value theorem") etc. 

Both the method of infinitesimal differentiation and infinitesimal integration was known to ancient Indian mathematicians.

The query was to bring a general awareness about the history of Calculus. My guide was a famous American mathematician, (late) Dr. Chris Byrnes, who got his doctorate from Dr. Marshall Stone (University of Massachusetts, Amherst, USA), and of course, Stone was known for his pioneering results in Algebraic Geometry and Analysis. (Late) Prof. Marshall Stone had visited India a few times.

We all can learn from each other!

http://www.math10.com/en/maths-history/math-history-in-india/hindu-maths/hindumaths.html

Dear Colleagues,

Good day,

Calculus, known in its early history as infinitesimal calculus, is a mathematical discipline focused on limits, functions, derivatives, integrals, and infinite series. Isaac Newton and Gottfried Leibniz independently invented calculus in the mid-17th century. However, each inventor claimed that the other one stole his work in a bitter dispute that raged until the end of their lives.

please, You could read further by following the attached links .

http://en.wikipedia.org/wiki/History_of_calculus

http://en.wikipedia.org/wiki/Leibniz%E2%80%93Newton_calculus_controversy

http://www.uiowa.edu/~c22m025c/history.html

Since ancient times, man has felt the need to count objects around him: for this he invented the numbers. Soon, the need to count has been joined to perform operations on numbers. Thus was born the arithmetic, the basic part of mathematics that deals with the study of numbers and operations that can be performed on them. The development of arithmetic has followed several milestones, which have led to the introduction of new numbers in addition to the natural numbers used to count: integer relative, rational numbers, real numbers

The term derives from the Greek word arithmetikè arithmetic, itself formed from the terms arithmos ("number") and tèchne ("technical"), and means "art of numbers." The arithmetic is, in fact, the art of combining the numbers according to different procedures, calls operations. It deals with the basic operations: addition (+), subtraction (-), multiplication (×) and division (:), fractions, extractions and root of logarithms.

The first to use the word 'arithmetic' were the disciples of Pythagoras. The Pythagoreans have established the distinction between even numbers and odd numbers and discovered the primes, ie those divisible only for the unit and for themselves. Around 300 BC Euclid, has exposed these and other concepts of arithmetic and geometry in his most famous work, the Elements. In later centuries have been interested arithmetic Archimedes and Eratosthenes, the inventor of the sieve, a method for finding prime numbers smaller than an entire assigned.

count

They are called natural numbers positive integers, ie 1, 2, 3, 4, ... The natural numbers were the first to be invented, driven by the need to count objects, and the act of counting, that is, to indicate the number of objects belonging to a certain set, started to mathematics.

In arithmetic, an operation is a process which combines two elements a third element, the result of the operation. The addition, the fundamental operation of arithmetic, is, in practice, the continuation of the act of counting. The continuity between the act of counting and arithmetic operations is reflected also in natural language, in which the expression arithmetic indicates precisely the execution of arithmetic operations.

What I have read in the classic book, Mens of mathematics by E. T. Bell , supports what Hazim wrote and you wrote in your introduction: Isaac Newton and Gottfried Leibniz independently invented calculus in the mid-17th century. However, each inventor claimed that the other one stole his work in a bitter dispute that raged until the end of their lives.

P.S. The last chapter of the book is about Srinivasa Ramanujan, a  genius who unfortunately died at the age of 33.

http://en.wikipedia.org/wiki/Men_of_Mathematics

As per available information, Sir Isaac Newton, an English physicist and mathematician, and Gottfried Wilhelm von Leibniz, a German mathematician and philosopher, are the forerunners for the title of the Father of Calculus. There are a few who believe Isaac Newton laid the foundation of this branch of mathematics, while others believe Leibniz discovered it first.

Hello Sundarapandian,

From the point of view of a person who attempted and failed differential equations three times I am not enamored with the first two and I guess now the domain becomes three. But in all seriousness from the point of view of a CST it is not surprising. The same is true in core classes for undergrates at the university I am with and is close to beiing a universal in the U.S. Two semesters of Western Civilazation, one of World history, history of Politics, and History of Social Theory. All Eurocentric, all required, and all math and science classes the debate you frame the question in has been solved. It is Newton today as it was in the 80's when I was in Jr. College

K-12 education is the same and in fact do to required class to graduate a student cannot breake away from this indocternation until a Jr. in college. To further this historical invisbleness Christopher Columbus is still the "discoverer" and a day off meanwhile the Native children children of the 12 Pubelo's their holowday's are marked as an unexcused absence.

So it is not surprising but samefull.

In solidarty

Douglas.

Different people that test the same hypothesis apparently accept or reject it with different degrees. Some people might, for instance, state that hypothesis ‘N’ is true with a probability of 75% whereas others do not believe hypothesis ‘N’ can work. Different people have different background information available definitely causing people to believe more or less in a given hypothesis. The scientific challenge then will be to identify reliability in individual-specific background information to judge claims from those that have more or less confidence in hypothesis ‘N’. Is it important to define historically interconnected chains in background information to ultimately support or reject a given hypothesis with higher or lower probability? Those that possess more historical information should be more efficient in hypothesis-testing when a scale of analysis is adequately defined.

Does it matter?

Dear Barry Turner - The history of any subject (here we are discussing Calculus) is just to motivate the students and it is not needed for scientists and researchers. As a teacher, I like knowing the history of the subject I teach, and historically how the results were discovered and by whom, any interesting stories, etc.

There is a very good book on Discrete Mathematics by Kenneth H. Rosen and this celebrated author has given historical account of almost all topics in Discrete Mathematics. It is a pleasure for people like me to teach a class with this book! 

Best regards.. Sundar

Dear Sundar

It matters little whether Leibnitz or Newton or Euler or any other person from antiquity 'invented' or 'discovered' calculus.  It is a great question for a pub quiz but has little merit elsewhere.  I love history as much as I do science but history and science are not sports.  Who cares who won the race?  It only matters that it was run.

Dear Barry, don't be such a wet towel. It matters, it matters to someone. It may matter in this sense: if calculus was developed at an earlier time, then the question arises, "why did it take so long to further develop it?", also, "What was the perception at that time to allow development of calculus?". 

Leben heißt kampf

What a wonderful motto, equating life with struggle. 

I am not normally a wet towel, I just could not give a monkey's toss who invented calculus.  I am grateful to them and I would happily buy them a beer but does it really matter?

I guess someone invented calculus, like someone invented sitting around doing nothing.  There are virtues in both.

Gan canny wi them speers man, divvent dunshus and Wah Geordies Man!

I've read that Newton and Leibnitz invented calculus. Newton have found calculus for practical reasons and Leibnitz worked on developing mathematical aparatus. Leibnitzs calculus notation is still used today. Before them Pierre de Fermat found some methods for finding tangents and extremes of some functions.

Those who down vote opinions they don't like are as reprehensible as the classroom cheat.  

Step forward into the light of open debate or be a sneak, snitch and a louse.  All good subjects for scientific study but not worthy participants in it.

Potential sources to alight within one the concepts of calculus abound. For example, consider drinking a beer, as one gulps down that beer, one may thinks to himself, "Hey, if only I could let fly me flap and pour, in one fell swoop, all mine beer in me gullet." Thus, the concept of discrete steps are born, then the limit to continuous differentials, finally, even Rolle's theorem, for obviously a beer is bounded, both from above and below, where one thinks, "Before I commence drinking, I have a full glass, but after the last gulp, alas, I have none; therefore, f(b)=f(a)=0, hence, f'(x)=0, where that differential lay is somewhere between the cackles of me woman and the ding of that voice in me heads".

Barry,

I agree with the down vote but I question your position on two fronts. Following your stance to its logical conclusion there would be no history. Meaning no answer to the question: Dad/mom were did this come from? Dad/mom what did people do before we had X, Y, Z. Dad/mom how did people.........? Without history we have no story, no grounding as to what and who we are. We are left with answering all those types of questions with a shrug of the shoulder, or "I-don-know". "what's it matter anyway", "who cares all that matters is now" and denies the richness of others past.

Second front, by denying the importance of Sundarapandian question demonstrates that in fact we still live in colonial times. It is a position that the conqueror takes in denying the very existence of personhood, cultures, civilizations. The heart and soul of othering, marginalization, dehumanism, racism, sexism, classism and any other "ism" one can think of including humanism. It is a tool by the dominant to continue the oppression and subjugation of the crown and flag. Dismissiveness allows the aftermath of the rape, pillage, plundering, genocide, torture, theft, and cruelty of the past to be labeled in the history books as the conquerors "the Age of Discovery".

The conqueror use of "who cares" allows the stereotyping of others to be validated at the same time denies the unavoidable responsibility for the human condition they left behind. Who cares is tool used in social reproduction, social stigma, and stagnation. Dismissal is a tatic used by colonizers as to avoiding answering the question why the extinction of all that once was of a group and culture. Dismissiveness also allows the conqueror to sleep at night in a bed and blanket in a house built on and by the suffering of millions of bleading, sore and dead backs.

However, dismissal is a double edge sword. On the one edge is the pain and misery inflected on others and the other edge? Illumination, as to the fact you know. In that knowledge the oppressed struggles, misery, starvation, lack of materiality comfort, pain and wealth rest assured that following the colonizers path leds to the slow suffocation of all that is left. The one precious gift the colonizers lost and what back is seen in the frustration of the torturer face as wip is drawn back. The one true thing of value that wealth, comfort, status and the disissmisal cannot get back? The humanity of the oppressed lost by the colonizers deveiled by the words of "who cares".

That is why it matters and besides Barry its quite frankly insulting.

I agree with the words "those who down vote opinions they don't like are as reprehensible as the classroom cheat". I take I take the "Step forward into the light of open debate or be a sneak, snitch and a louse. All good subjects for scientific study but not worthy participants in it" and you can bet your last shilling on that.

Douglas

Douglas

Indeed it is the principle of science and the doctrine of research to search for truth and know exactly where things came from. Not only young people who asks that question but every research scientist. It is therefore a good question to ask  where did first calculus came to be entertained as a subject.

When we teach calculus, the first thing we do is to stimulate students by telling what is calculus ? where did it come from? who discovered calculus and for what reasons? These are basic and part of the course topics we always deal with. But as Dear Sundarapandian pointed out, it is a legitimate and scientific quest to make as to who indeed first discovered calculus?  

I have read that Bhaskara, an Indian person who practiced mathematics and astrology was using calculus for his works more ahead of Newton and Leibniz did. This is a good finding which has to be put in text books of calculus so that truth is thought to young students and for the sake of science.

It is not only calculus that interests me,  the Diophantine equation with an identifying name:  Pell's equation was also an equation well studied by an Indian person named Brahamgupta, well far far before John Pell, the person who owned the equation. But the name was attached to him by Euler for a wrong identification.

There are such mistakes happened in history for wrong reasons I assume not by intention, but when discovered it has to be amended and be part of  true history of the subject.  

Chinese mathematicians have contributed immensely in several areas.

The enclosed link gives a bird's eye-view of the great contributions of Chinese Mathematics.

http://en.wikipedia.org/wiki/Chinese_mathematics

How many brilliant scientists reinvented the wheel? If people cannot read Chinese, how can they exploit historical facts that are presented in the Chinese literature?

And how often information remains hidden in private reports or reports that cannot be made public for X reasons?

Two British researchers challenged the conventional history of mathematics having evidence that the infinite series, one of the core concepts of calculus, was first developed by Indian mathematicians in the 14th century. They also believe they can show how the advancement may have been passed along to Isaac Newton and Gottfried Wilhelm Leibniz, who are credited with independently developing the concept some 250 years later. Historians have long known about the work of the Keralese mathematician Madhava and his followers, but Joseph says that no one has yet firmly established how the work of Indian scholars concerning the infinite series might have directly influenced mathematicians like Newton and Leibniz. At the International Conference on Indian History, Civilisation and Geopolitics (ICIH 2009) held at New Delhi in January, Dr. C.K. Raju presented a paper about how calculus, an Indian invention, was picked up by the Jesuit priests from Kerala in the second half of the 16th century and taken to Europe. This is how the Westerners got their calculus. Over time, people forgot this link and the Europeans began to claim calculus as their own invention. This myth still persists despite calculus texts existing in India since thousands of years. The contributions of India to mathematics have been tragically overlooked. Excerpted from the following threads:

http://www.cbc.ca/news/technology/calculus-created-in-india-250-years-before-newton-study-1.632433

http://indianrealist.com/2009/01/26/how-jesuits-took-calculus-from-india-to-europe/

Newton and Leibnitz are two of the greatest thinkers in history but the argument over who 'discovered' calculus diminishes both.  We do give accolades to those who discover and invent but we should avoid attaching significance beyond that.  Newton was a spectacularly brilliant man but he was also a flawed one.  The biggest problem with the 'who discovered/invented first' question is that it often leads to chauvinism and triumphalism.  

There are a number of threads on RG where a peculiar attachment to who did what first seems to eclipse the actual discovery and there are some where supremacy is even hinted at.  Some people think that inventions and discoveries can be attributed to the nationality of the scientist.  Newton was English, Leibniz was Saxon.  It is doubtful that in the 17th century that nationality mattered to either of them. 

Congratulations to scientific greats of all countries and all times but lets not get carried away with any significance of nationality or race.  Humans are wonderfully innovative but it is their humanity that make them so, not their national origins or allegiances.

Barry,

Why use the term race? There is no such a thing! It is a socially constructed false concept meant to justify the inhumane treatment because differant humans were claimed not to human human (the science of eugenics).. Where I live my ansesters called the original people savages and stole 30,000 acers of farm land under the illusion of the homestead act.

No Barry I am a man of low melanin have no national attachments. I am also a man that many upon sight associate me with colonizers and rightfully so given the history.

Being in that place demands challenging any hint of colonizer language as payment do to my brothers and sisters of the community of humanity.

Douglas 

"Who decovered ...?" and similar questions are the jobs of historians. If any result of such a research gives reason that one nation feels better than another, such historian research is misguided.

Personally I admire all great scientists which produced results and scientific progress and I don´t mind to which continent, nation or "outfit" they belong.  Affinity to what ever of these characterisations is no reason to develop to a great researcher.

So I totally agree with @Barry Turner´s last comment.

Douglas

I agree entirely that race and nationality are artificial constructs, that is entirely my point. I have noticed a tendency on these threads to equate the idea with who invented what with concomitant comments such as the comment above regarding the contribution of 'India' to mathematics as being overlooked.  

India is a state invented by humans in 1949 and it has not contributed anything to mathematics.  That some brilliant mathematicians emanated from within its boundaries is irrelevant just as it is irrelevant that many of history's great engineers were Scottish.  

We should avoid associating a persons nationality with scientific achievements.  That a person is British, Mexican or Mongolian has nothing to do with their scientific excellence but some may interpret it that way.  That is only a stones throw from the idea of master races.

Science should not be associated with tribalism, race or sectarianism and nationality is only a form of those less edifying features of humanity.  It might be Newton that invented calculus, it could be Leibniz, it could equally be an Ancient Egyptian, a Greek or a Mayan or Inca.  In fact it could be anybody.  The sad fact to date is that the argument nearly always comes down to was it an Englishman?  or a German (Saxon actually)? or perhaps an Indian?  That is why I asked... who cares?  

Geometric shapes have just been found carved on a 450,000 year old shell.  It could not have been Homo Sapien that did that, they were not around.  It was a hominid however and it is the oldest deliberately made mark ever found.  The Mapasangat pebble was almost certainly collected by an Australopithecine nearly 3 million years ago because its naturally occurring feature resemble a face.  

We should wonder at such things because a person did them, not because they may have lived in a land that is now a nation.  I agree totally with what you say.  There is no such thing as race or nationality in any real scientific sense.  Both are constructs to justify supremacy, ill treatment of others and chauvinism.  A plague on all nations!

Barry Turner wrote: "India is a state invented by humans in 1949 and it has not contributed anything to mathematics."

Barry also adds in the next line: "That some brilliant mathematicians emanated from within its boundaries is irrelevant just as it is irrelevant that many of history's great engineers were Scottish."

I am deeply offended by both comments.

India got its independence from British rule in 1947, and in the past, India was known as "Bharat", though it was divided into various small kingdoms.

Whether Barry gives credit or not, the world of Mathematicians gives credit to the contribution of Indian Mathematics to the world.

India invented 'zero'. (Hope Barry does not say -  "So India's contribution is  just zero!!")

So many research articles have come on various mathematical fields. Ramanujan has contributed wonderful results in Number Theory, which have various applications in Cryptography etc. There are many others also working in Analysis who have published wonderful results. In Applied Mathematics, especially Modelling, Indians have contributed a lot. In Economics and Statistics also, there are great contributions from Indian Mathematicians. If I have to prepare a list, it will be a very loooooooong list!

I have given Wikipedia reference on Indian Mathematics, but this is a bird's eye-view only, and not a comprehensive list. There is a lot more to add.

In the second link, Wikipedia says, "Indian mathematicians have made a number of contributions to mathematics that have significantly influenced scientists and mathematicians in the modern era. These include place-value arithmetical notations, the ruler, the concept of zero, and most importantly, the Arabic-Hindu numerals predominantly used today and which can be used in the future also."

In the list of Indian mathematicians provided by Wikipedia, one has to just click the Mathematician's name to see the contributions.!!

http://en.wikipedia.org/wiki/Indian_mathematics

http://en.wikipedia.org/wiki/List_of_Indian_mathematicians

Calculus is invented by Mathematicians from all of the world; Calculus contains hundreds of concepts and each of these concepts took tens of years to be developed, defined, and redefined and to take at the end its final form as we see these days. see for example the definition of one concept: the LIMIT at the link:

http://en.wikipedia.org/wiki/Limit_%28mathematics%29

Calculus is not one number, it is a complex infinity that not one person could claim an invention of it. It is accumulation of many and multiple thoughts.

I think Mr Barry is defending the wrongs of the past by being simply dismissive and using a carpeting down  word "who cares" but at the same time keeping the false history of things to be thought. It is true that the development of scientific concepts are outcomes of activities of almost all human beings starting from antiquity and transferred from one place to another through different means and therefore it is very difficult to definitively say this person discovers this.

Newton said, I see beyond what others can not see because I stand on the shoulders of giants. That simply means he got every thing from an existing knowledge and from his predecessors but developed them.  If we again search where the Indian person got his ideas of calculus or anything we are discussing about, then it will lead us to an other rout and different place. Thus to declare Newton or this guy or that guy single handedly  created this subject or concept is I think a flowed testament to make in the first place. 

Besides  the British past had a voracious appetite of simply taking every thing to them, giving their titles and names of their kings to every new thing and new place they encountered eliminating the original names :  ... king island, ...James territory, ...Victoria mountain , ---- Wales river, .... country, ----king street, ---Bell's equation, Pell's formula, -...subject, .... ocean, etc.  These are facts of the past we know and this thread points one particular case of the past. 

For instance since antiquity to the mid 19 century the southern part of Atlantic ocean was called Ethiopian ocean or Ocean Aethiopica ( see http://www.zazzle.com/antique_old_world_map_of_africa_c_1635_poster-228277013083183806 ,   :http://en.wikipedia.org/wiki/Aethiopian_Sea , http://www.raremaps.com/gallery/detail/37149/Africae_nova_Tabula_1631/Hondius.html) but some how some one who had the privilege to publish modern maps removed  the name Ethiopian ocean and instead put the  name Atlantic ocean. This is simply eliminating history by being dismissive with I don't care mind set but at the same time promoting self.  

Hello all and good evening Barry,

Barry for your mental and ohysical health, get some help.

Douglas

Please, at which level are we comunicating?

Sorry and I to you Barry I also Apologize, I was beging and hit add before I was even started, I send this out before I begin.

To all my mistake 

Douglas

I ask a simple question here.  Are the mathematicians that came from India brilliant because they are mathematicians or because they came from India?  

Was James Clerk Maxwell a brilliant mathematician because he was Scottish?

Alan Turing was one of the most brilliant mathematicians of the 20th century.  He was British but his brilliance was not anything to do with where he came from and the disgraceful treatment he received at the bidding of an ungrateful nation testifies to that.  

I would ask all rational people to consider the implications of focusing on nationality when considering scientific or mathematical excellence.  Nationalism and chauvinism is  not something to be proud of.  An Individuals excellence or for that matter repugnance is not related to the passport they carry.  It certainly is not related to their ethnic origins, religious or philosophical beliefs or politics.  

I find it very disturbing that when assessing a person's contribution to science and mathematics that some find their national origins to be considered important.  I have no doubt that many significant mathematicians came from India as they did from every other country in the world.  That is of no more significance than how tall they are or the football team they support.

Ethical science transcends the brutal and ignorant world of nationalism and we do not have to look very far back in history to see where science in the name of a so called superior culture took us.  It was not my intention to be offensive and I fail to see how questioning the relevance of nationality to mathematics could, in any way be offensive.  

I am fully aware that my 'own country' occupied and exploited its empire for greed, aggrandisement and a sense of superiority and supremacy.  The British Empire is certainly nothing to be proud of to most educated English people and they are appalled by it and its legacy.  That is why so many educated people reject nationalism and look forward to its extinction.

Oh! one more thing I don't need any help.  

Dear Barry - Thanks for your clarification. I like to address only this issue.

You wrote - "I find it very disturbing that when assessing a person's contribution to science and mathematics that some find their national origins to be considered important."

The question "Who really discovered Calculus" is a historical query. We are discussing names only - Newton, Leibniz and Bhaskara II - and as this is a historical query, the nationality of these great mathematicians are also mentioned. (I hope you agree Bhaskara II was a great mathematician, but if you debate on it, it is also fine).

No-one who answered in this thread attempted in any way in assessing Newton's contributions or Leibniz's contributions per se. The subject of the query was only about Calculus, and Newton, Leibniz and Bhaskara had contributed in other fields also. No comparison was made about them. In my description of the query itself, I wrote that Newton and Leibniz discovered "Calculus" independently and using different routes as well (Differential Calculus for Newton and Integral Calculus for Leibniz). Without actually defining what is "d/dx" (the derviative), Bhaskara II had conceptualized it many centuries ago, got the formula that "d(sin x) /dx = cos x" and in addition to this, Bhaskara II proved a theorem, which is now known as "Mean Value Theorem". Even in India, Bhaskara's contributions are not taught in the schools, as most of us follow western text-books only on Calculus.

It is a general query about different viewpoints on the history of Calculus, and some have contributed academically in this thread. I have also highlighted the contribution of Chinese Mathematicians from a historical perspective.

I completed my doctoral program under the guidance of (Late) Prof. Chris Byrnes, who was a famous Control Professor from USA. Chris Byrnes was a doctoral student of a great mathematician named Marshall Stone (USA), who in turn got his PhD degree under the guidance of George D. Birkhoff (USA). My guide was very proud of his American heritage, and also the contributions of Stone and Birkhoff. He shared his lineage details to me only so as to get inspired, and not feel let down by any failures in research. The historical details are important & useful for inspiration.  All countries have produced some great people in Academics & Research, and ResearchGate is a good forum where we can all learn from each other.

Regarding my research guide, Dr. Byrnes, famous American Professor, I have great regards for him. Some of his students even called him by his first name "Chris"! I was very shocked to know his sudden demise. He is always a great role model for me.

In the memory of all those 600 who marched, suffered, and died for at the bridge in Selma, Alabama on March 7, 1965. "Bloody Sunday"

Hello all,

I would just like to add my last comment on this matter. Sundarapandian, I want to thank you on the question you pose, it is a great question for several reasons. 1) That mathematics itself being a global historical language. 2) Demonstrates the powerful connection between history and cultural/national identity. 3) Can also flush out hidden agendas and Ideologies. (see attachment)

I entered Sundarapandian's question because it was intriguing. I had never heard of Bhaskara II, and had taken Calculus 1,2 ,and 3. I followed the question and Mr. Turner's post of "does it matter" puzzled me. I questioned myself as to why and my answer was it must of brought up an issue or hit a personal nerve. The second question was twofold; Why not just post and move on and if not why not? In continuing to follow a pattern emerged in Mr. Tuner's postings that I read about 25 years ago. Sense then I have seen repeated for years but felt I had to wait and make sure at some point it would become obvious to as many as possible regarding Mr. Turner's continued particaption. I felt that moment came when Mr. Turner posted his reply to Mr. Cundin's post. The motive was made even clearer in Mr. Turner's outburst to being "down voted" and that is when I began responding in kind. I also see a similar pattern (not included in attachment) in the down votes cast. This points to the hypocrisy between Mr. Turner's contention of stepping into the light of debate by Mr. Turner himself or those who agree with Mr. Turners point. I happen to agree with Mr. Turner on this point, I also agree that a reason other than I did not like it accompany the down vote because I would like to read and learn or open a dialogue concerning the issues raised, or throw it where it belongs.

Mr. Turner my apogee was sincere in my post only in the fact that it was not contextualized. Mr. Turner you do need help. Whether or not you get it is up to you. It has been clearly evident to me that your comments have been a conscious act designed to be hurtful, veiled in your use of "nationalistic arguments" which are at best shallow and transparent. Your behavior of not addressing respectfully any person of color as to their views having value, demeaning Mr. Vaidyanathan question as merely a "pub question", and your name calling assault on these fine men is an expression of views. Mr. Turner you had no idea who down voted your response you merely assumed it had to be one of these fine men because it could not be a British subject or a Saxon (as you continually point out in many posts). Hence I not only believe but know these posts are to be assigned to a narrow minded Neanderthalic mind which is an expression of ideological racism. That mind set did not allow the concept of a man such as myself to exist. Yes, I do exist, I am a man who the minute I see racism at work, smell it, touch it, I do as you challenged "Step forward into the light of debate" as if that ideology pose any such thing as debate. Nor does anyone who holds such a belief pose much intellect or intelligence enough to be a challenge in a debate. The tiny little ideological view belongs back in the 19th century where at least that belief, for a time, had the "science of eugenics" to support racist beliefs. If anyone need empirical data to the claim I make find thread or ask the people who's question on RG used the term "retarded" in describing my human brothers and sisters as to their intellectual challenges as to how to teach them.

Mr. Turner, your "It was not my intention to be offensive and I fail to see how questioning the relevance of nationality to mathematics could, in any way be offensive" post again is a conscious act of inflecting pain on others. Nobody could have made this point more obvious to an "educated British subject" than Sundarapandian himself , in his post. The sentence is also a reflection as to how deeply believed by racist the falsehood of biologically different "races". Those who hold and act from the ideology of racism are a disgrace to all that is, love, understanding, compassion, respect, and knowledge that do transcend science and are made a mockery of by such people when they are uttered.

At the end of the day all those who believe such; I feel nothing but pity, disgust, and contempt towards because just their physical body alone associates them with what is a man of science, a man, or could ever be possibly be considered to be a man is really the antithesis of.

I also understand that by bringing to light racism and racist acts, although being true, may have consequences. Whatever those may be are more than likely the same as other institutions have and I have paid and am willing to pay again. They have and always will to honor the memory of Jimmy Lee Jackson who was mortally wounded in February of 1965 and in contempt for the man who has never been to trial and by his own words admitted to it with same spoken pattern I recogonized and that sparked my opening to this response.

The man I describe above and who's pattern I recognized and my apologue was directed to was you Mr. Turner. My apologue in my previous post was only due to the fact it was not contextualized. Mr. Turner you do need help because your conscious or unconscious beliefs were like the classic 1939 movie The Wizard of Oz hidden behind the drape which has now been pulled away and have been exposed. Whether or not you get it is up to you. As far as stepping into the light is this good enough. If not personally e-mail me at [email protected] I know many good counselors and physiologists I have worked with for 20 years ago I was you.  

Douglas

Suggesting that a person 'needs help' simply because you do not agree with them is hardly the language expected of rational debate.  It is not so much an insult to me, since that hardly matters.  It is however a slur on these who suffer mental health problems as it suggests that any opinions they may have can be disregarded on the grounds that they 'need help'

I have worked with mentally ill people for over 30 years and train those who care directly for them.  I have in those 30 years observed that often far more sense comes out of those labelled 'mentally ill' than it does out of some who mistakenly believe their opinions rationally held.   

It is indeed a fact that those who hold and act from the ideology of racism are a disgrace.  The basis of all prejudice is ignorance and all racists, like all who discriminate against the mentally ill live in that dark world.  I need no help in recognising that. 

Oh and by the way, Leibnitz was from Saxony.  The state now known as Germany did not exist when he was around so it would be quite inaccurate to describe him as German.   That is what I mean when I criticised ascribing nationalities to human endeavour, it is clearly flawed.  Nation states like ideologies are transient. Leibniz would not have identified himself as German and Bhaskara would not have described himself as coming from a country that did not exist in the 14th century.  

We in the UK attach all kinds of significance to the 'nationality' of our historical heroes.  Queen Boudicca is held in high standing here for instance.  It would however be quite absurd to describe her as English or British.  She lived 2000 years ago and although the part of the country she came from is now part of the UK such a political entity was not known at the time.  

Returning to Leibnitz.  He was born in Leipzig, a city I am very fond of and remember from its days in the German Democratic Republic.  Leibnitz was not however a citizen of that state, any more than a citizen of the Federal Republic of Germany  since neither came into being until 1949.  Anyone born in Leipzig since 1991 is also not a citizen of the GDR since it ceased to exist in 1990.

It is always helpful to differentiate (not in the sense of calculus of course) between nation states and race.  Nation states are an irrelevant and temporary political manifestation that cause most of the worlds hatreds, divisions and woes but there is only one human race, Homo Sapien Sapien.  

Mr. Turner,

Mr. Turner my reply to you sir was based on your own words, never acknowledging or addressing a person of color's posts, never accepting or respecting Sundarapandian words of deep offence, the hollowness of your arguments, ignoring how hurtful your argument was and persisting to use it, and the insistence on being right.

Sir if I had wanted to offend you I surely would have. But take heart, you did stop the discussion. I would assume for a racist point of view that would be a victory. When I deal with people who's shoes are the same, I do my best to point out the facts, expect three stages of responses and I make a decision,

1) Denial

2) Combativeness

3) The "but this" or "but that" or "but my point is....."

4) If stuck here after years of experience working with men's groups on racism and racist acts I move on.

If they reach stage four of self reflection I stay.

Good by Mr. Turner.

Remember the words of Carter Woodson (1933) “as another has well said, to handicap a student by teaching him that his black (brown) face is a curse and his struggle to change his condition is hopeless is the worst sort of lynching”.

And “If you can control a man's thinking you do not have to worry about his action. When you determine what a man shall think you do not have to concern yourself about what he will do. If you make a man feel that he is inferior, you do not have to compel him to accept an inferior status, for he will seek it himself. If you make a man think that he is justly an outcast, you do not have to order him to the back door. He will go without being told; and if there is no back door, his very nature will demand one.”

Peace be to all,

Douglas

I hope someone brings the discussion back into track!!

Who really discovered Calculus??

Muchas Gracias!

Dear Sundarapandian,

If I may, it seems we have two competing theories of who discovered calculus. One the one hand by one person at one time (Mr. Tripathi). On the other  hand  an accumulation over time by many people (Mr. Rababah and Mr Kilani).

If I may ask when you use the word "discover" what is its meaning? I know we all know what the word means but applying to calculus I get the sense we may possibly have different interpretations.

Therefore, discover = calculus is ? Meaning from your point of view the basic proposition of calculus. Is it a single equation leading to others? A  new conseptual paradigm of mathematics?  Please I now this would be helpful to me.

Peace,

Douglas 

Coming back to the point and looking to the timeline of Calculus discovery, it appears that Indian scholars have actually laid down the foundation of Calculus. In 1021, Ibn al-Haytham developed and proved the earliest general formula for infinitesimal and integral calculus using mathematical induction. During 12th century Bhāskara II conceives differential calculus, and also develops Rolle's Theorem, Pell's equation, a proof for the Pythagorean Theorem, computes π to 5 decimal places, and calculates the time taken for the earth to orbit the sun to 9 decimal places. Madhava, considered as the father of mathematical analysis, in 14th century, worked on the power series for p and for sine and cosine functions, and along with other Kerala school mathematicians, founded the important concepts of Calculus. During the same period another scholar, Parameshvara, a Kerala school mathematician, presents a series form of the sine function that is equivalent to its Taylor series expansion, states the mean value theorem of differential calculus, and is also the first mathematician to give the radius of circle with inscribed cyclic quadrilateral. Again, Madhava discovered the series expansion for the inverse-tangent function, the infinite series for arctan and sin, and many methods for calculating the circumference of the circle, and uses them to compute π correct to 11 decimal places. In 1501, Nilakantha Somayaji wrote the ‘Tantra Samgraha’, which laid the foundation for a complete system of fluxions (derivatives), and expands on concepts from his previous text, the ‘Aryabhatiya Bhasya’. Further, in 1550 Jyeshtadeva, a Kerala school mathematician, wrote the ‘Yuktibhāṣā’, the world's first calculus text, which gives detailed derivations of many calculus theorems and formulae. In view of all these historical facts, it is amply clear that seed of Calculus were germinated in India and saplings were further spread in the west. However, the contribution of Indian mathematicians remained unnoticed and unrecorded. Origin of calculus need to be critically relooked.

It is often the case that those with extensive experience of combatting racism are induced you to see it everywhere, including in a debate about the history of calculus. Once again the nationality of its discoverer is irrelevant and nationality is nothing to do with race. It is always best to try and keep emotion out of scientific or intellectual debate, it is a distraction.

While it is a fascinating historical question in the big scheme of things it is not significant. Calculus is a tool and like other tools it was discovered time and time again by people all over the globe. Fire and hand-axes are a good analogy. This technology was discovered by many many individuals right across the world and without it there would be no calculus and most likely the human race would have been extinct by now.

The hand-axe and fire of course were both discovered by Homo Erectus or Homo Ergaster and they were of course a different species to modern man.  Modern man enjoys the fruits of their labour and intuition just as we enjoy the great discoveries of all humanity.  

One thing is always certain no one discovered anything without the efforts of those who went before them as Yogesh acknowledges above when describing the influence Pythagoras and Ibn al-Haytham had on Bhaskara.  Science and mathematics do not spring into the world fully formed, they all take generations to develop and calculus is a case in point.  The concept is written nanos gigantum humeris insidentes or, as Isaac Newton said in a rare expression of modesty.  "I have seen further because I stand on the shoulders of giants".  

The early 13th century mathematician Leonardo of Pisa known to the modern world as Fibonacci wrote the treatise “Liber Abaci” in which he introduced Hindu-Arabic mathematical concepts to Europe and made the inefficient system of Roman numerals obsolete.  Leonardo came from a city that is now part of Italy but he would not of course have recognised such a state and it would be inaccurate to call him Italian. 

Leonardo is today perhaps most famous for his mathematical model based on the breeding of rabbits and called the Fibonacci Sequence. That sequence was first recorded by mathematicians living in what is now India in the 6th century. The contribution of mathematicians from what is now described as the Indian Sub-Continent is well documented as is that of the Ancient Greeks and Mesopotamians.

The saddest fact of all about the history of mathematics is that such a marvelous endeavor has been held back by ideology, politics and imperialism. Leonardo of Pisa travelled extensively in North Africa where he learned from Islamic mathematical scholars. This quest was of course often frustrated by the activities of those fighting Crusades and Jihads for the purpose of establishing supremacy of one culture over another.

Just imagine how much more knowledge and skill would have been disseminated and so much earlier if greed, ambition and mindless ideology had not set humanity against itself in the name of tribe, religion, nation and empire.

Dear Mr. Tripathi,

If I may given the beauty of the time line you have provided what concepts or methods would you or any consider to have been added to what is now modern day calculus? So we may complete the journey?

Peace,

Douglas

Sundarapandian,

Quotes from https://en.wikipedia.org/wiki/Calculus:

‘’ Calculations of volume and area, one goal of integral calculus, can be found in the Egyptian Moscow papyrus (c. 1820 BC), but the formulas are simple instructions, with no indication as to method, and some of them lack major components.[5] From the age of Greek mathematics, Eudoxus (c. 408–355 BC) used the method of exhaustion, which foreshadows the concept of the limit, to calculate areas and volumes, whileArchimedes (c. 287–212 BC) developed this idea further, inventing heuristics which resemble the methods of integral calculus.’’’

In Europe, the foundational work was a treatise due to Bonaventura Cavalieri (1598 –1647), who argued that volumes and areas should be computed as the sums of the volumes and areas of infinitesimally thin cross-sections. The ideas were similar to Archimedes' in The Method, but this treatise is believed to have been lost in the 13th century, and was only rediscovered in the early 20th century, and so would have been unknown to Cavalieri. 

‘’  Pierre de Fermat, claiming that he borrowed from Diophantus, introduced the concept of adequality, which represented equality up to an infinitesimal error term.[12] The combination was achieved by John Wallis, Isaac Barrow, and James Gregory, the latter two proving the second fundamental theorem of calculusaround 1670.’’

‘’ The product rule and chain rule, the notion of higher derivatives, Taylor series, and analytical functions were introduced by Isaac Newton in an idiosyncratic notation which he used to solve problems of mathematical physics.’’

‘’ These ideas were arranged into a true calculus of infinitesimals by Gottfried Wilhelm Leibniz, who was originally accused ofplagiarism by Newton.[14] He is now regarded as an independent inventor of and contributor to calculus. His contribution was to provide a clear set of rules for working with infinitesimal quantities, allowing the computation of second and higher derivatives, and providing the product rule and chain rule, in their differential and integral forms. Unlike Newton, Leibniz paid a lot of attention to the formalism, often spending days determining appropriate symbols for concepts.’’

‘’ Newton derived his results first (later to be published in his Method of Fluxions), but Leibniz published his Nova Methodus pro Maximis et Minimis first. Newton claimed Leibniz stole ideas from his unpublished notes, which Newton had shared with a few members of the Royal Society. This controversy divided English-speaking mathematicians from continental European mathematicians for many years, to the detriment of English mathematics. A careful examination of the papers of Leibniz and Newton shows that they arrived at their results independently, with Leibniz starting first with integration and Newton with differentiati