14th March, declared as the International Pi day, is an interesting date, particularly when expressed in the way the Americans describe the dates - 3/14. A closer look at the date (month and day) reveals a combination of numbers that represent an approximation that constitutes the commonly used constant value for π (Pi) - 3.14. Coincidentally, this date also happens to be the birth date of one of the all time great scientists, Albert Einstein, who was born on the 14th March, 1879, in Germany. In recognition of this 14th March is celebrated as the international Pi day. Pi is perhaps one of few symbols, which has evoked extraordinary human curiosity, mystery, romanticism, misconception and interest, the evidence of which goes back in time.
Since historic times one of the profound challenges faced by mathematicians has been a precise calculation of the ratio between a circle's circumference and its diameter - which has come to be known by the Greek letter π (Pi) - whose history is as old as human desire to measure. However, the symbol for this ratio, known today as π (Pi), dates from the early eighteenth century. It was in the year 1706, a little-known mathematics teacher, named William Jones, first used the symbol π to represent an ideal that in numerical terms can be approached, but never reached. The value π (Pi) is a constant that we get when we divide any circle’s circumference by its diameter and its value is approximately 3.14. The beauty of this enduring mathematical constant lies in the fact that if one were to keep calculating π’s digits with more and more accuracy, you will discover that the fraction goes on and on, literally forever, with no predictable or periodic pattern to the fractions. It is this unpredictable nature of the unending fractions that makes π (Pi) a special constant, which has interested mathematicians for several centuries.
From ancient India, Babylonia and to the Middle Ages in Europe to the present day supercomputers, the mathematicians have been constantly striving to calculate this mysterious constant Pi. They have tried to search for exact fractions, formulas, and, more recently, patterns in the long string of numbers that start with 3.14159 2653........ to represent the unending value of this enchanting mathematical constant - an irrational number. The discovery of this interesting mathematical constant Pi (π), has unending applications in various fields of science and engineering. Notwithstanding the fact that there perhaps are quite a number of tall and illogical claims of ancient Indian achievements in the fields of aviation, cloning etc., which have been confused with mythology that are despised, rightly so, by the likes of Nobelists Venky Ramakrishnan and others, we must also admit that India did contribute immensely in the field of science and technology with discoveries and inventions in certain areas of science, mathematics, arts and culture. Of the numerous areas in which India has made immense contributions one of the unparalleled contributions is the discovery of the value of Pi, which dates back to the period of Sulbha Sutra, and more precisely to Aryabhata’s times.
Pi - the ratio of the circumference to the diameter - was of extraordinary significance for ancient Indians, primarily because of the exacting standards of measurements that were necessary for the construction of the sacred fire altars for performing religious ceremonies, primary among them the हवन. One of the practices that existed in India during the early times (so called Vedic times) was that each of the households were supposed to have three different shaped fire altars in the form of a circle, square and a semicircle, for performing the sacred हवन. The primary conditions of these three different shaped fire altars was that their areas had be equal. This perhaps necessitated a precious measurement and calculation of the value of Pi. Professor Ramasubramanian, Department of Humanities and Social Sciences, IIT Bombay, is an expert in this field and has so thoughtfully explained the rich history of calculation of Pi in India in a video, the link for which is appended at the end of this article. He says the necessity to construct equal area fire altars in the shape of square, circle and semicircle warranted a precise measurement and discovery of the value of Pi. However, it is said that Archimedes was the first to suggest a specific number for the value of Pi, which he represented as 22/7, a value of which is continuing to be used even in modern days.
Several Indian mathematicians, including eminent mathematician and astronomer Aryabhatta (476- 550AD) worked on the problem for attempting to find a precise value of Pi. Primary among them was Aryabhata, who worked on the approximation value for Pi (π) and his value 3.1416 is far more accurate as opposed to the value 22/7, which was suggested by Archimedes. Aryabhata also came to the conclusion that Pi is irrational. In the second part of the Aryabhatiya (gaṇitapāda 10), he writes : “चतुरधिकम् शतम् अष्टगुणम्द्वाषष्टि तथा सहस्राणाम् अयुत द्वय विष्कम्भस्य आसन्न वृत्त परिणाह” (caturadhikam satam ashtagunam dvāṣaṣṭi tatha sahasrāṇām Ayutadvayavi kambhasyasanno vṛttapariṇāhaḥ), meaning ; “Add four to 100, multiply by eight, and then add 62,000. By this rule the circumference of a circle with a diameter of 20,000 can be approached.” When you calculate this value, the answer comes to 3.1416, which is very close to the modern value of Pi to the fourth fraction (3.14159). You will also notice that Aryabhata very interestingly uses the word आसन्न āsanna (approaching / approximating), to mean that the resultant value of the ratio of circumference to the diameter of the circle ( Pi) is an approximation. He further adds that the value is incommensurable (or irrational). Aryabhata’s seminal work ‘Aryabhatiya’ provides an insight into his works.
Aryabhatiya is a compendium of mathematics and astronomy, which has survived till modern times. It reveals beyond doubt that Aryabhata had indeed discovered and worked on concept of Pi long before the Western world was even aware of its existence. The Aryabhatiya, written in Sanskrit consists of 108 verses divided into 4 padas or chapters. The second pada called the Ganitapada (Ganita, meaning mathematics) bears a reference to the concept (and approximate value) of Pi. Considering the period when Aryabhata calculated the value of Pi, around 499 AD, one can definitively state that he had a sophisticated insight into this enchanting constant. The profoundness of his insight can best be understood when we look at the period when the irrationality of Pi (π) was proved in modern times by Johann Heinrich Lambert, in 1761 almost 1300 years later. The works of Aryabhatiya was translated into Arabic (some time during 820 AD) and this approximation was mentioned in Al-Khwarizmi‘s book on algebra, through which it reached Europe.
The oldest recognised representation of infinite series for the value of Pi, is now ascribed to Gregory and Leibniz. But then almost three hundred years before Gregory and Leibniz’s formulae came to the fore, there was a series which was codified in the typical Indian traditions and form of a verse by one of the greatest mathematicians of a India - Madhava from the Kerala School of Mathematics. Mādhava of Sangamagrāma (1340 - 1425), a renowned mathematician and astronomer from Kerala is also considered to be the founder of the Kerala school of astronomy and mathematics. Madhava was one of the greatest scholars of mathematics that Kerala produced during the medieval period. He was the first to use infinite series approximations for a range of trigonometric functions, which is hailed by many as a profound contribution in moving from a finite series to treat their limit-passage to infinity. Madhava also made pioneering contributions to the study of infinite series, calculus, trigonometry, geometry, and algebra. Some scholars have also suggested that Madhava's work, through the writings of the Kerala school, may have been transmitted to Europe via Jesuit missionaries and traders who were active around the ancient port of Muziris at the time. As a result, it may have had an influence on later European developments in analysis and calculus.
Madhava was followed by equally famed mathematicians in Kerala, primary among them include Parameshwara and, Neelakantha, in whose works citations are available to establish Madhava’s marvelous achievements in mathematics. Some of these are the values of π correct to 10 places of decimals, imposing corrections to infinite series after certain terms for quick and better results, derivation of Sine & Cosine power series for computing better Sine and Versed- sine tables, which are unique by contemporary standard. In an article published in the Indian Journal,of History of Science, an INSA publication Dr A K Bag says ‘Madhava’s value, as quoted by by Neelakantha and Shankara, is far better than others and considerably much closer approximation. And he adds that from one of the works of Shankara an evidence is provided to Madhava’s more precise measurement for the value of Pi. In four verses (atra–ha ma–dhava), Shankara says that Madhava had actually suggested a method for finding the circumference of a circle by means of constructing a number of regular polygons, for the sum of the sides of the polygon, which would almost be equal to the length of the circumference of the circle. Step by step procedure was adopted to compute the side of a square- polygon for a circle, then half-side of the square- polygon (octagon), then half-side of the octagon (hexadecagon), then half side of the hexadecagon (32- gon) and so on indicating that the number of regular polygons had to be large for considerably accurate value. Thus suggesting a solution for an infinite series. Madhava’s π series was later rediscovered, about 250 years later, in Europe by scholars —Wilhelm Leibniz (1673), Newton (1675), De Lagney (1682), De Moivre (c.1720), Euler (1748) and others.
Indian mathematicians interest in the value of Pi continued into the nineteenth and early twentieth century. The great Indian mathematician, Ramanujan, about whom I have written a blog, had a great obsession for Pi and this obsession would follow him until his last. His note books contain hundreds of different ways of calculating the approximate values of Pi. In just the two notebooks he wrote, before arriving at Cambridge, are found 400 pages of formulas and theorems. Courtesy the theoretical foundations that Ramanujan laid a century ago, powerful computers can now calculate the first 10 trillion decimals of the number Pi.
Such has been an intense interest in the value of Pi by not just Indian mathematicians but also globally, that it is no wonder that the approximate value of Pi - 3.14 - which is mostly used in ordinary calculations, has been declared as the international Pi day. The Pi day has an interesting connect with the science centres across the world, particularly in the US. Founded in 1988 at the Exploratorium - the pioneer in the evolution of interactive science centres in the world including the science centres in India (NCSM) - the Pi (π) Day has become an international commemorative event that is widely celebrated all around the world, and my colleague Mr Ramachandran, Director of the Mother museums of NCSM, Birla and Industrial and Technological Museum, Kolkata sent us an image of their celebrations of this day with the students in Kolkata, which is also appended in the article.
Wishing you all a very happy international Pi Day and may the excellence of achievements of Indians in the field of mathematics, continue to inspire generations to come.
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