Mathematics

Calculus & India

Print edition : January 23, 2015

The statue of Aryabhata on the Inter-University Centre for Astronomy and Astrophysics campus in Pune.

There is probably no hope of ever coming to a firm conclusion on whether calculus originated in India. But if one considers the question whether or not Indian mathematics witnessed modest beginnings in calculus, then it appears that both sides of the argument have some merits.

THERE are some topics that create a buzz in the country from time to time and heat up the atmosphere, putting global warming to shame. Did the ancient Indians know quantum mechanics? Do the animal faces of certain gods mean that the ancient Indians practised plastic surgery? Did they come up with calculus?

Some topics are outright nonsense, but there are some that merit a debate. However, most of us are not historians, and such discussions become pointless after a while. One has come across the names Aryabhata or Varahamihira in school textbooks, but one would be hard pressed to name any specific achievement of these ancient Indian scientists. School students are taught about the discovery of the zero but they never learn what it means to have “discovered” it.

How does one suddenly invoke a number and bring it down to the realm of reality? We teach our children that our ancestors must have had a remarkable knowledge of metallurgy and give the example of the iron pillar in Delhi, but we never spell out what exactly they knew.

This superficiality in our collective knowledge often leads to meaningless, rhetorical debates on the achievements of ancient Indian scientists. Some people believe that the Vedas contained everything worth knowing. And there are others who contend that Indians were a barbaric lot before the Europeans arrived. Often, these debates take on a political flavour. By saying something in favour of ancient Indian scientists, one runs the risk of being called a “Hindutva” aficionado, while one might be called unpatriotic or irreverent of one’s heritage if one chooses to criticise such a position.

Recently, there were claims in the media that the important mathematical tool calculus, instrumental in the advancement of modern science, originated in India much before Isaac Newton or Gottfried Wilhem von Leibniz, who are usually credited with its development. If the claim turns out to be true, then much of what is known about the history of Indian science will have to be revised. But there has hardly been any discussion in the popular media outlining the details of the arguments for or against the proposition. The apprehension that one would be pigeonholed into either the Hindutva camp or called unpatriotic stalks much of these discussions.

Another reason for skirting the details is that there are as yet no black-and-white answers to this question. The best answer probably lies between a complete yes or no. One might think that this is a diplomatic answer, but the real story behind the claim of calculus having originated in India is as complicated as it is fascinating.

One could begin by explaining what calculus means. In simple words, calculus allows one to study the rate of change of certain parameters or values. Consider the fluctuations in the stock market index. Suppose one was told how the index varied during a month, then one could easily find the average rate of change in the index over the course of a month by considering its value at the beginning and at the end of the month. But the index may not have changed in a uniform manner over the month. There is no reason why it should. Someone might be interested in knowing how it varied over short timescales, such as days or weeks. This information of the rate of its change over short time intervals may be important for some purpose. And calculus is the appropriate tool for such a task.

Consider another example, say, the orbit of an artificial satellite. It has an elliptical orbit and, therefore, comes closest to the earth at some point of time and then reaches the farthest point in its orbit at a later time. The satellite will move at a high speed when it comes close to the earth and will slow down when it reaches the farthest point. Its speed, therefore, changes all the time. Calculus can be of help if one wishes to find out how its speed changes with its location or to measure its speed at every moment, not the average speed but the instantaneous speed of the satellite as it occurs at every instant. That is where calculus can become a powerful tool of mathematics.

Aryabhata

Ancient Indian mathematicians and astronomers were indeed curious about the instantaneous rate of change of certain orbital parameters. Almost a millennium before Newton, one Indian mathematician started wondering about this instantaneous rate of change and thought it would be a worthwhile topic to pursue. Aryabhata, in the sixth century, noticed something curious in his table of sines of angles. He not only came up with the definition of sine that is the closest to what is used today but also created the most sought-after ready reckoner for sines in that era.

There is an interesting story about his sine tables. The term “sine” has come to us in a torturous way from Aryabhata and is a linguistic curiosity. The story of the word sine is a perfect example of the meaning of the phrase “lost in translation”.

Aryabhata had defined sine in the following way. Consider a circle of radius unity (1 unit). Draw a chord within the circle so that the two ends of the chord subtend an angle at the centre. The sine of half of that angle will be half the length of the chord.

The calculation of different orbital parameters involves the measurement of sines of various angles, and it is useful to have a ready-made table in which sines of different angles are tabulated. One can then use the table to look up sine values instead of calculating them from scratch. Such tables were in high demand among ancient and medieval astronomers. And Aryabhata’s sine table was considered the most accurate among them. He had tabulated the sines of 24 angles, equally spaced between zero and 90 degrees (with a difference of 3.75° between them). The table was in the form of a verse, and he had come up with an ingenious way of coding the table in the words of the poem. The introduction to his book Aryabhatia explained the code.He had given the name jya for the sine of an angle since it means “chord” in Sanskrit. When Muhammad ibn Musa al-Khwarizmi translated his work in the ninth century, for some reason he wrote jiva instead of jya. Much later, in the 12th century, when Gerard of Cremona translated it into Latin, he thought it was a typo and “corrected” it to the Arabic word jaib, which means a fold or a pocket. He translated it into Latin as sinus, which means a bend or a hole made by a fold. It was later Anglicised to the word sine, which is used today.

This story clearly shows how important Aryabhata’s sine table was in those days. But he did not rest with creating a table for others’ use. Being a perceptive mathematician, he noticed an oddity in the values of the sines of two consecutive angles in the table. Consider a table of the sines of angles such as 1°, 2°, 3°, and so on. He took the difference between the sines of two consecutive angles, for example, 2° and 3°, and then made a column of the differences. Then he made yet another column for the second level of differences between these first column of differences. In other words, he calculated how the difference between the sines of 3° and 2° differed from those between 4° and 3°, and so on. (Aryabhata’s table had angles spaced by 3¾°, but the calculations of differences were similar.) Aryabhata noticed that the second-level differences were proportional to the sines in the first column! To put it in a different way, he first measured how rapidly the sines of angles increased with increasing angle. Then he measured how this difference itself changed from small to large angles. Aryabhata had found that these second level of differences were proportional to the sine itself. This will be familiar to any high school student as it is a known result of modern calculus that the second “derivative” of sine is proportional to sine. Aryabhata did have an inkling of this result on the basis of his observations of his sine table.

This does not prove that Aryabhata had realised the potential of calculus. He did notice a curious connection between the double differences of sines of angles and the sines themselves, but he had not thought of it as the “rate of change” of sine. He had also not thought about what use such calculations might have and what paths they could lead to. Consider once more the example of the share index. Suppose the average index during January is 20,000 and it increases to 21,000 in February and to 23,000 in March. (Real stock markets, thankfully, do not behave like this; if they did readers of this article would stop reading and rush to buy stocks.) Then, one would have said that the average monthly index increased by 1,000 a month from January to February, and then by 2,000 a month from February to March. One would then get an idea of how the monthly average varied from month to month. But it would still be an average rate of change of the index.

Suppose that one now wished to estimate the value of the index on a particular day (between January and March, say February 10) on the basis of this rather sketchy information, one could only do this in an approximate manner. If one wished to improve upon one’s “prediction” (more like “post-diction” in this case), then one would need better information on how the index varied, not from month to month but from week to week or, even better, on a daily basis.

This is the crux of calculus. The finer the interval over which the averaging is done, the better one can estimate the instantaneous rates. How fine an interval does one need for this purpose? The answer is limited only by one’s imagination. In practice, one tries to find the values of a parameter (stock market index, in this case) as closely spaced as possible, in mathematical terms, one would say, as the spacing tends to zero. Mathematicians call this the limiting value of a parameter.

Brahmagupta

A century after Aryabhata, another mathematician became interested in the differences in the sine table and probably realised what a gold mine it was. Brahmagupta was an exceptional mathematician although his notions in astronomy were as conservative as his ideas in mathematics were extraordinary and adventurous. He took Aryabhata to task for thinking that the earth moved around the sun and thought little of Aryabhata’s concern about the precession of equinoxes. His views ultimately led to a faulty calendar system that is still in use in this country. But when it came to mathematics, his contributions and insights would put him among the best mathematicians in the world of all times.

Almost 35 years after writing his magnum opus, Brahmasphuta-siddhanta, in which he spelled out the definition of zero, he started thinking about how to take forward Aryabhata’s insight into the differences in the sines of angles.

Suppose one has the information on the rate of change of the share market index with a spacing of, say, a few days. Then, one could use the value of the index on a particular day and estimate—with the help of the information on its rate of change —the index on an intermediate date for which one had no previous information. It is like jumping from one tree to another in a forest. Mathematicians call it interpolation. But as in the case of jumping between trees, the success of this method depends on the distance between the trees, or how closely the data are sampled. If the “distance” is too large, the method will fall flat on the ground. If one wanted to estimate the value of the index on, say, February 10, then it would be wise to use the datum for a day that comes closest to February 10, and not for a day much earlier, say, in January, as the estimate would have a large margin of error.

Mathematicians before Brahmagupta did have an idea of estimating the sines of the intermediate angles that did not figure in Aryabhata’s table, using the sine of an angle that came closest to the required one and the information of the sine differences. But the method failed if the angle was not close enough to one of the entries. Brahmagupta came up with a formula for interpolation using the second differences, which allowed him to take leaps in angles as large as 15°. This method (known in modern mathematics as Stirling’s formula) is routinely used today, and it was clearly a remarkable achievement. This was the first interpolation formula in the history of mathematics that used two levels of differences in data (or, in mathematical terms, a second-order interpolation). It is not clear how Brahmagupta derived the formula—he never explained it anywhere—but historians think he used geometry.

However, Brahmagupta worked with differences and not really the instantaneous rate of change of parameters. Then, a few centuries later came Bhaskara II. He was the first person to say that Brahmagupta’s interpolation formula could be interpreted in terms of an instantaneous rate of change. He used Brahmagupta’s interpolation formula in the case of planetary orbits to estimate the tatkalik (Sanskrit for instantaneous) orbital speed of celestial objects. This was indeed a giant leap in thinking. Bhaskara II (who, incidentally, was born exactly 1,000 years ago) went on to enunciate the methods of finding the maximum and minimum speeds, which he said, corresponded to the derivative being zero.

Madhava

This is where the matter stood for a long time. One can imagine that political unrest in northern India at this time contributed to the decline or, at least, lack of further progress on this subject. A few centuries later, there was a resurgence of mathematical studies way down south, near the backwaters of Kerala. A mathematician, Madhava, who lived near present-day Thrissur, came up with a formula for the sine of an angle that involved the fourth derivative, or the fourth level of differences in the sine table. It is clear that such a formula will always be approximate, and therefore have an infinite number of terms, each smaller than the preceding one and each providing a steadily diminishing correction to the value of the sine. The further one proceeds into the series of these terms, the better is one’s estimate although at each step the correction keeps decreasing, as the estimate keeps approaching the actual value.

Like his predecessor Brahmagupta, Madhava never explained how he came up with the formula. Historians of mathematics again suspect that he obtained it through geometrical methods. This series for the sine of an angle is very close to the series in use today, and this formula comes from the methods of calculus. But there is a crucial difference between Madhava’s series and the one derived from calculus. All the terms in the formula are weighed by a fraction (a coefficient) that determines how small (or large) the contribution of that particular term is to the total sum. The fractions in the modern series run as 1, ½, 1/6, and so on, whereas in Madhava’s case, the third term had a coefficient of ¼ instead of 1/6. One might think this is hair-splitting, but historians have written papers on the significance of this small difference. For them, it is a sign that Madhava did not know the methods of modern calculus.

They, therefore, argue that Madhava’s achievement does not necessarily mean that ancient or medieval Indian mathematicians were anywhere close to developing what is known as calculus today. Madhava and his followers probably had a vague idea at best of what they were doing but never realised its potential. For one thing, Indian mathematicians were only interested in the rates of change of trigonometric parameters, whereas as a part of modern mathematics, calculus is used as a tool to discuss the rate of change of anything that varies. For example, the value of sine changes when the angle is varied. In modern terms, it is called a function of the angle. When the angle, the variable, changes, the value of the function also changes. Modern calculus gives one the methods of finding the rate of change of functions when the variable changes. In other words, modern calculus was developed not only with trigonometry in mind but also in a more general manner.

Others contend that even the pioneers of modern calculus were sleepwalkers, to borrow the phrase that Arthur Koestler used for early astronomers such as Johannes Kepler. It has been pointed out that Newton, Leibnitz and others also did not quite fathom the full potential of the remarkable tool that they had developed. The initial practitioners of modern calculus also never fully explained how they derived their results. The language of discussions had not developed enough and was not sophisticated enough to describe what they were doing.

A few Indian historians have gone further and suggested that perhaps early European mathematicians had got the idea of calculus from Indian sources. In the 16th century, Jesuit priests became interested in Indian mathematical and astronomical manuscripts because of their application to navigation. Christopher Clavius was one such person. He had reformed the mathematical syllabus at Collegio Romano, and one of his students, Matteo Ricci, had come to Cochin (Kochi) and written to him that he wanted to learn Indian methods of timekeeping “from an intelligent Brahman or an honest Moor”. Some historians argue that this proves that there was a flow of information from East to West through Jesuit priests. It is not inconceivable, they argue, that Madhava’s ideas (written in his Yuktibhasa in Malayalam, a language that the Jesuits knew well by that time —by 1590, they were even teaching Malayalam in schools in Cochin) had found their way to Europe and seeded the development of calculus there.

The other camp claims that such an extraordinary claim must have solid evidence. No written document from Europe mentions any Indian source, they point out. Historians from the first camp feel that this requirement of firm evidence is too stringent and probably smacks of a colonial attitude.

This is where the debate stands. There is probably no hope of ever coming to any firm conclusion unless some new manuscript comes to light and proves or disproves one of the claims. If one, however, considers the question whether or not Indian mathematics witnessed modest beginnings in calculus, then it appears that arguments on both sides have some merits.

It is true that Brahmagupta, Bhaskara and Madhava did not use the methods of modern calculus. At best, they were probably groping in the dark. But it is also true that they did have a sense of the instantaneous rate of change of parameters although their ideas could not be pinned down with rigour because they did not have a clear notion of limits of functions. There were certainly some seeds of what is called calculus today, but it appears that these did not fully germinate and bear fruit as happened later in Europe.

Biman Nath is an astronomer, a science writer and a novelist. He is currently a professor of astronomy at the Raman Research Institute, Bangalore.

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