There are certain interesting numbers in Mathematics like Pi, e, etc. The numbers basically started with natural numbers one to infinity of course infinity was not even conceived as a number and so too was zero so in the initial phases and therefore we start with natural numbers and then we move on to whole numbers, then we move on to integers, the negative numbers, that itself is an interesting history on its own, anyway, The point that I wanted to make is this number π(pi) which we refer to, to represent the ratio of circumference to the diameter has a very interesting history. Why were they interested in calculating the value of Pi to high degree? One way to see is, it has to do with the area of a circle and you cannot avoid it, but then its nature was never understood. Infact there are books which describe it as the most mysterious number. And today we have understood that to be a transcendental number. It was first recognized as irrational number, and irrationality itself is not so easy to understand, so people struggled for thousands of years, and to have an exact representation of Pi, so since it is irrational, it may not be possible for you unless you have an infinite series in the right hand side. So any series that you have, which adds up, if it is a terminating series, so it’ll turn into a rational number. So unless you have infinite series, there are various infinite
series representations for Pi. One interesting series which is today ascribed to Gregory – Leibniz, is a series which was codified in the form of verse almost 300 years before
Gregory – Leibniz by a mathematician called Madhava, as we understand today, Madhava was in 14th Century, so these people have been in 17th century. So historically this is something which is not so well known. And if the series were to be given a name which actually honours the founder, then it should be really called Madhava series, instead of Gregory–Leibniz series, this is a historical note about Pi. What was the context in which people wanted to know the value of Pi in the Indian tradition? If we look back, the kind of altars which
people have been using thousands and thousands of years before, for performing various rituals, every household was suppposed to have 3 altars, one in the form of a circle, the other in the form of square and the third in the form of semi circle. So these were called as “Tretaagni”. One is called “Aahavaneeya”, “Dakshina” and “Garhapatya”. The constraint that people imposed, was, that the three altars should have the same area. Now you can very easily see as to why they wanted to know the value of Pi. So it occurs in that context. So if the circle area as we understand today is “pi r squared” and that should be equated to “a squared” and the area should be same. and we have some sort of approximation in the Shulba Sutras. Shulba sutras as I told you is around 800 BC, we do not know exactly, because there is no reference which is found in Shulba Sutras, which gives us a hint to make an estimate on the date of composition as we find for instance in Aryabhatiya or
later works. Aryabhata for instance gives a verse where in he clearly states, Shashtyabdaanam shashtyardaa vyateethaaH trayascha yugapaadaaH TryadhikaavimshatirabdaaH tadehamamajanmanoteetaaH So he very cleary says the year of composition. and after Shulba sutras, we find a nice approximation for Pi which has been given by Aryabhata. Aryabhata’s verse goes like this Chaturadhikam shatamashtaguNam dwashashTistatha sahasraaNaam | Ayutadwayavishkambhasya aasanno vrittapariNaahaH || This is a very interesting verse. Interesting in the sense that it gives you a very clear picture that it has been clearly recognized by Aryabhata that it is only approximate value. So how do we come to know, because he has used the word “aasanna”. “asanna” means that which is close by. aasannaH vrittapariNaahaH – how do they express? So they express the circumference on the left hand side in terms of diameter multiplied by some quantity. This is how it goes. Chaturadhikam shatam ashtaguNam. So he basically gives you a ratio. So circumference should be mentioned in terms of diameter. So here he says, if this were the diameter, then, you have to multiple by this term to get the circumference. ashtaguNa means multipled by 8. Chaturadhikam shatam – 104. So 4 above hundred . dwashashTistatha sahasraaNaam. sahasraaNaam dwashashTiH. dwashashTiH is 62. So Sixty two thousand. Sixty two thousand plus this So it comes to 62,832. Ayutadwayavishkambhasya. Vishkambha is the term used for Diameter. Vistaara, Vishkambha, Vyaasa, these are the terms that they generally use for diameter. Ayutam is 10,000. Ayutadwaya is 20,000. aasanno vrittapariNaahaH. vrittapariNaahaH – circumference of a circle. So circumference of a circle is close to this. What does this boil to? So this boils to getting a value which is 3.1416 as the value of Pi. So this is correct to 4 decimal places. Infact people have been asking, so how did Aryabhata get this value, which is correct to 4 decimal places. So it is not something which one can draw and then keep measuring by something, some scale by which you will be able to get this kind of accuracy, it is almost impossible. So they had a way of doubling methods, Circumference doubling, you divide that into various parts and you can nicely get some formulae, which essentially involves computing square root and square. So if you have a way, technique to find the square root correctly and the square if you have algorithms for that, then you will keep on increasing your accuracy by taking more and more divisions on it. So Polygon Doubling method seems to have been the method which should have been used by Aryabhata to get this value.