Indian Contribution to Mathematics Does no one remember the Hindu contribution to Mathematics? The link to the article

Johnee disagree with the title of this thread and what you are trying to prove, its more of indic maths than hindu. Religion has got to do nothing with the knowledge, invention or anything. Its the region where it came from. Arabs didn't grow a brain after they became muslims. The mesopotamian civilization is one of the earliest one and it contributed a lot to human development. So did the sumerian. They gave us the wheel. Now where does religion come into the picture? by the title of the thread then since a lot of invention in modern era has happened in countries in the west which are majority christian, then do we term all of them as christian this and christian that? In short I don't agree to the religious connotation to knowledge.

Yusuf, Its perogative to agree or disagree. Its not religious connotation to knowledge but simply a reminder that a certain religious group has made contributions to a field. Knowlege is universal, no denying that. But different groups contribute to it and their contributions need to be credited in history.

I would go more as indic contribution than a religious based. What if I were a hindu all my life and then converted to islam and one fine day eureka, I invent something? So what who is to be credited? Me being from india or a follower of a faith?

BTW, its not my title, its the original title given by the author. Indian mathematicians in the past were of the Dharma-based traditions. And it is right that this is made clear today, lest there be any confusion

Well then I disagree with the author mate. For some reason it may be to do with trying to create an identity. Or credit or what. I am reminded of how every time an american of indian origin or an indian in any other country gets successful or wins an election or something is quickly caught by indians especially the media who go ga ga about a person of indian origin reaching the zenith. That person may be a third gen indian there who may have never visited india, but indians don't tire of trying to "own" them.

A hypothetic situation can at best only generate clueless guesses. Anyway, the point that needs notice is that in west the early scientists in quest of knowledge had to defy existing traditions and were witch hunted in the process. While, in contrast, the ancient scientists in India were aided(and not obstructed) by the existing traditions of Hinduism. Science and religion gelled well in ancient India as compared to west(or even modern India). This pattern needs notice and appreciation given the violent persecution of the early scientists(like Galileo) in the west.

You are right about the media and the general attitude. But I dont see any connection of that with the present topic. I dont understand why you are against the credit being given where its due. Are you against Hindus being given the credit of the work they have done in mathematical(and other scientific field)? If so why? As for creating identities, no new identity is being created. The identity already exists. Their positive contribution is being appreciated which has been unfairly credited to others. For eg: mostly arabs are being credited with the present decimal number system we use. But we know that its actually hindus who invented it.

Yusuf, the above article shows the hindu background(or vedic background to be more precise) was not just an accident but infact a delibrate contributor to the achievements.

Johnee I don think attaching mathematical inventions with religion is appropriate. Lets, say can we say the modern science is 'Christian science'?? the credit for those inventions should be given to the civilization, to be precise, to the guy who invented it despite his religion.

Xebex, read the last posted article. That article shows how the ancient indian contributions were directly related to the tradition. And thats the reason for this thread. It needs noted and appreciated that the ancient Indian traditions were not obstructive to the development of science but in contrast directly encouraging. You know that the development of modern science has started in west but it was not encouraged by the existing traditions, instead they did everything within their power to stifle it. So, while the ancient indian science and its achievements can be credited to the existing religions and the traditions the same is not the case with the western(or modern) science. Hope, you understand the difference and dont use a broad brush for all things. Religion and science are not anti to each other, they have never been in India, this syndrome has been mostly a hallmark of west or western born traditions or religions.

Aryabhata Aryabhata stands as a pioneer of the revival of Indian mathematics, and the so called 'classical period', or 'Golden era' of Indian mathematics. Arguably the Classical period continued until the 12th century, although in some respects it was over before Aryabhata's death following a costly, if ultimately successful, war with invading Huns which resulted in the eroding of the Gupta culture. Some of his contributions are: * Tables of sine values * The Aryabhatiya is the first historical work of the dated type, which definitely uses some of these (trigonometric) functions and contains a table of sines. * Of the mathematics contained within the Aryabhatiya the most remarkable is an approximation for π (pi), which is surprisingly accurate. The value given is: π = 3.1416 * In the field of 'pure' mathematics his most significant contribution was his solution to the indeterminate equation: ax - by = c Aryabhata's Table of Sign vales shown in "Hindu Sine" Column The Aryabhatiya was translated into Arabic by Abu'l Hassan al-Ahwazi (before 1000 AD) as Zij al-Arjabhar and it is partly through this translation that Indian computational and mathematical methods were introduced to the Arabs, which will have had a significant effect on the forward progress made by mathematics. The historian A Cajori even goes as far as to suggest that: ...Diophantus, the father of Greek algebra, got the first algebraic knowledge from India. [RG4, P 12] Link

Bhaskaracharya II Bhaskara II, is regarded almost without question as the greatest Hindu mathematician of all time and his contribution to not just Indian, but world mathematics is undeniable. As L Gurjar states: ...Because of his work India gave a definite 'quota' to the forward world march of the science. [LG, P 104] He wrote Siddhanta Siromani in 1150 AD, which contained four sections: * Lilavati (arithmetic) * Bijaganita (algebra) * Goladhyaya (sphere/celestial globe) * Grahaganita (mathematics of the planets) Bhaskara is to be the first to show that: deltasin x = cos x deltax Bhaskara was fully acquainted with the principle of differential calculus, and that his researches were in no way inferior to Newton's, asides the fact that it seems he did not understand the utility of his researches, and thus historians of mathematics generally neglect his outstanding achievement, which is extremely regrettable. Bhaskara also goes deeper into the 'differential calculus' and suggests the differential coefficient vanishes at an extremum value of the function, indicating knowledge of the concept of 'infinitesimals'. He also gives the (now) well known results for sin(a + b) and sin(a - b). There is also evidence of an early form of Rolle's theorem if f(a) = f(b) = 0 then f '(x) = 0 for some x with a <>, in Bhaskara's work. Lilavati: Definitions In Lilavati, the beautiful definitions of different units demonostrates the advanced arithmetics of the Vedic-era. 1. Having bowed to [Ganesa] who causes the joy of those who worship him, who, when thought of, removes obstacles, the elephant-headed one whose feet are honored by multitudes of gods, I state the arithmetical rules of true computation, the beautiful Lilavati, clear and providing enjoyment to the wise by its concise, charming and pure quarter-verses. 2. Two times ten varatakas [cowrie] are a kakini [shell], and four of those are a pana [copper coin]. Sixteen of those are considered here [to be] a dramma [coin, "drachma"], and so sixteen drammas are a niska [gold coin]. 3. Two yavas [barley grain (a weight measure)] are here considered equal to a gunja [berry]; three gunjas are a valla [wheat grain] and eight of those are a dharana [rice grain]. Two of those are a gadyanaka, so a ghataka is defined [to be] equal to fourteen vallas. 4. Those who understand weights call half of ten gunjas a masa [bean], and sixteen of [the weights] called masa a karsa, and four karsas a pala. A karsa of gold is known as a suvarna [lit. "gold"]. 5. An angula [digit] is eight yavodaras [thick part of a barley grain]; a hasta [hand] is four times six angulas. Here, a danda [rod] is four hastas, and a krosa [cry] is two thousand of those. 6. A yojana is four krosas. Likewise, ten karas [hand, hasta] are a vamsa [bamboo]; a nivartana is a field bounded by four sides of twenty vamsas [each]. 7. A twelve-edged [solid] with width, length, and height measured by one hasta is called a cubic hasta. In the case of grain and so forth, a measure [equal to] a cubic hasta is called in treatises a "Magadha kharika". 8. And a drona [bucket] is a sixteenth part of a khari; an adhaka is a fourth part of a drona. Here, a prastha is a fourth part of an adhaka; by earlier [authorities], a kudava is defined [as] one-fourth of a prastha. [For more Dept of Mathematics, Brown Univercity] Ancient Scripts in Lilavati Link

Madhava of Sangamagramma Born in Cochin on the Kerala coast Madhava of Sangamagramma (c. 1340 - 1425) is one of the greatest mathematician-astronomer of medieval India. Sadly most of his mathematical works are currently lost, although it is possible extant work may yet be 'unearthed'. His most significant contribution was in moving on from the finite procedures of ancient mathematics to 'treat their limit passage to infinity', which is considered to be the essence of modern classical analysis. Although there is not complete certainty it is thought Madhava was responsible for the discovery of all of the following results: 1. theta = tan theta - (tan3 theta)/3 + (tan5theta)/5 - ... , 2. rtheta= {r(rsintheta)/1(rcostheta)}-{r(rsintheta)3/3(rcostheta)3}+{r(rsintheta)5/5(rcostheta)5}- ... 3. sintheta = theta - theta3/3! + theta5/5! - .. Madhava-Newton power series. 4. costheta = 1 - theta2/2! + theta4/4! - ..., Madhava-Newton power series. 5. p/4 approx 1 - 1/3 + 1/5 - ... plusminus 1/n plusminus (-fi(n+1)), i = 1,2,3, and where f1 = n/2, f2 = (n/2)/(n2 + 1) and f3n/2)2 + 1)/((n/2)(n2 + 4 + 1))2 6. pd approx 2d + 4d/(22 - 1) - 4d/(42 - 1) + ... plusminus 4d/(n2 + 1), etc, this resulted in improved approximations of p, a further term was added to the above expression, allowing Madhava to calculate p to 13 decimal places. The value p = 3.14159265359 is unique to Kerala and is not found in any other mathematical literature. A value correct to 17 decimal places (3.14155265358979324) is found in the work Sadratnamala. R Gupta attributes calculation of this value to Madhava, (so perhaps he wrote this work, although this is pure conjecture). 7. tan -1x = x - x3/3 + x5/5 - ..., Madhava-Gregory series, power series for inverse tangent, 8. sin(x + h) approx sin x + (h/r)cos x - (h2/2r2)sin x 9. It is interesting that a four-term approximation formula for the sine function so close to the Taylor series approximation was known in India more than two centuries before the Taylor series expansion was discovered by Gregory about 1668. G Joseph states: ...We may consider Madhava to have been the founder of mathematical analysis. Some of his discoveries in this field show him to have possessed extraordinary intuition. [GJ, P 293] With regards to Keralese contributions as a whole, M Baron writes (in D Almeida, J John and A Zadorozhnyy): ...Some of the results achieved in connection with numerical integration by means of infinite series anticipate developments in Western Europe by several centuries. [DA/JJ/AZ1, P 79] Link

Possible Transmission of Indian Mathematics to Europe There is a very recent paper (written by D Almeida, J John and A Zadorozhnyy) of great interest, which goes as far as to suggest Indian mathematics may have been transmitted to Europe. It is true that India was in continuous contact with China, Arabia, and at the turn of the 16th century, Europe, thus transmission might well have been possible. However the current theory is that Indian calculus remained localised until its discovery by Charles Whish in the late 19th century. There is no evidence of direct transmission by way of relevant manuscripts but there is evidence of methodological similarities, communication routes and a suitable chronology for transmission. A key development of pre-calculus Europe, that of generalisation on the basis of induction, has deep methodological similarities with the corresponding Kerala development (200 years before). There is further evidence that John Wallis (1665) gave a recurrence relation and proof of Pythagoras theorem exactly as Bhaskara II did. The only way European scholars at this time could have been aware of the work of Bhaskara would have been through Indian 'routes'. The need for greater calendar accuracy and inadequacies in sea navigation techniques are thought to have led Europeans to seek knowledge from their colonies throughout the 16th and 17th centuries. The requirements of calendar reform were imperative with the dating of Easter proving extremely problematic, by the 16th century the European 'Julian' calendar was becoming so inaccurate that without correction Easter would eventually take place in summer! There were significant financial rewards for 'anyone' who could 'assist' in the improvement of navigation techniques. It is thought 'information' was sought from India in particular due to the influence of 11th century Arabic translations of earlier Indian navigational methods. vents also suggest it is quite possible that Jesuits (Christian missionaries) in Kerala were 'encouraged' to acquire mathematical knowledge while there. It is feasible that these observations are mere coincidence but if indeed it is true that transmission of ideas and results between Europe and Kerala occurred, then the 'role' of later Indian mathematics is even more important than previously thought. Link

Examples of Vedic-age Algebra 1. Use Bhaskara’s method to find two integers such that the square of their sum plus the cube of their sum equals twice the sum of their cubes. (This is a problem from Chapter 7 of the Vija Ganita.) 2. Show that the formula given by Brahmagupta for the area of a quadrilateral is correct if and only if the quadrilateral can be inscribed in a circle. Link

Examples of Vedic-age Trigonometry 1. Show that Aryabhata’s list of sine differences can be interpreted in our language as the table whose nth entry is Use a computer to generate this table for n = 1,2,...24 and compare the result with Aryabhata’s table. 2. If the recursive procedure described by Aryabhata is followed faithfully (as a computer can do), the result is the following sequence. 225, 224, 222, 219, 215, 210, 204, 197, 189, 181, 172, 162, 151, 140, 128, 115, 102, 88, 74, 60, 45, 30, 15, 0 Compare this list with Aryabhata’s list, and note the systematic divergence. These differences should be approximately 225 times the cosine of the appropriate angle. That is What does that fact suggest about the source of the systematic errors in the recursive procedure described by Aryabhata? Answer. The differences form the sequence 0, 0, 0, 0, 0, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 4, 4, 5, 5, 5, 6, 7, 7, 7. What we have here is an algebraic function attempting to approximate a transcendental function and losing track of it. If Aryabhata did use this formula, he must have started over with a new “seed” several times in order to avoid the error accumulation shown here. 3. Use Aryabhata’s procedure to compute the altitude of the Sun above the horizon in London (latitude 51 degree 32min) at 10:00AM on the vernal equinox. Assume that the sun rises at 6:00 AM on that day and sets at 6:00 PM. Link