Before dipping into the vast ocean of mathematics, let us take some deep breath and visit our country’s rich heritage of mathematics. You can be proud to know the fact that this branch of science took birth in this country.In all early ages of civilizations, the first expression of mathematical understanding appears in the form of counting systems. Number Systems in early societies were typically represented by groups of lines, though later different numbers came to be assigned specific numeral names and symbols (as in India) or were designated by alphabetic letters (such as in Rome).
Mathematical Activity in the Vedic Period
In the Vedic period, proceedings of mathematical activity are mostly to be found in Vedic texts associated with ritual activities. Arithmetic operations (Ganit) such as addition, subtraction, multiplication, fractions, squares, cubes and roots are enumerated in the Vishnu Purana by Ved Vyas (pre-1000 BC).Pythagoras - the Greek mathematician and philosopher who lived in the 6th C B.C was familiar with the Upanishads and learnt his basic geometry from the Sulva Sutras. An early statement of what is commonly known as the Pythagoras theorem is to be found in Baudhayana's Sutra - The chord which is stretched across the diagonal of a square produces an area of double the size. His Sutra also contains geometric solutions of a linear equation in a single unknown. Examples of quadratic equations also appear.Philosophical formulations concerning Shunya - i.e. emptiness or the void may have facilitated in the introduction of the concept of zero. While the zero (bindu) as an empty place holder in the place-value numeral system appears much earlier, algebraic definitions of the zero and it's relationship to mathematical functions appear in the mathematical treatises of Brahmagupta in the 7th C AD. Tangible evidence for the use of the zero begins to proliferate towards the end of the Gupta period.
The Indian Numeral System
Although the Chinese were also using a decimal based counting system, the Chinese lacked a formal notational system that had the abstraction and elegance of the Indian notational system, and it was the Indian notational system that reached the Western world through the Arabs and has now been accepted as universal. Several factors contributed to this development whose significance is perhaps best stated by French mathematician, Laplace: "The ingenious method of expressing every possible number using a set of ten symbols (each symbol having a place value and an absolute value) emerged in India. The idea seems so simple nowadays that its significance and profound importance is no longer appreciated. It's simplicity lies in the way it facilitated calculation and placed arithmetic foremost amongst useful inventions."
Influence of Trade and Commerce, Importance of Astronomy
The growth of trade and commerce, particularly lending and borrowing demanded an understanding of both simple and compound interest which probably stimulated the interest in arithmetic and geometric series. Brahmagupta's description of negative numbers as debts and positive numbers as fortunes points to a link between trade and mathematical study. Knowledge of astronomy - particularly knowledge of the tides and the stars was of great import to trading communities who crossed oceans or deserts at night. The science of astronomy was also spurred by the need to have accurate calendars and a better understanding of climate and rainfall patterns for timely sowing and choice of crops.The world did not have to wait for the Europeans to awake from their long intellectual slumber to see the development of advanced mathematical techniques. India achieved its own scientific renaissance of sorts during its classical era, beginning roughly one thousand years before the European Renaissance. Probably the most celebrated Indian mathematicians belonging to this period was Aaryabhata, who was born in 476 CE.In 499, when he was only 23 years old, Aaryabhata wrote his Aaryabhatiiya, a text covering both astronomy and mathematics. With regard to the former, the text is notable for its awareness of the relativity of motion. This awareness led to the astonishing suggestion that it is the Earth that rotates the Sun. He argued for the diurnal rotation of the earth, as an alternate theory to the rotation of the fixed stars and sun around the earth. He made this suggestion approximately one thousand years before Copernicus, evidently independently, reached the same conclusion.With regard to mathematics, one of Aaryabhata's greatest contributions was the calculation of sine tables, which no doubt was of great use for his astronomical calculations. He also developed methods of solving quadratic and indeterminate equations using fractions. He calculated pi to four decimal places, i.e., 3.1416. In addition, Aaryabhat.a invented a unique method of recording numbers which required perfect understanding of zero and the place-value system.Bhaskar I continued where Aryabhatta left off, and discussed in further detail topics such as the longitudes of the planets; conjunctions of the planets with each other and with bright stars; risings and settings of the planets; and the lunar crescent. Again, these studies required still more advanced mathematics and Bhaskar I expanded on the trigonometric equations provided by Aryabhatta, and like Aryabhatta correctly assessed pi to be an irrational number. Amongst his most important contributions was his formula for calculating the sine function which was 99% accurate. He also did pioneering work on indeterminate equations and considered for the first time quadrilaterals with all the four sides unequal and none of the opposite sides parallel.
Applied Mathematics, Solutions to Practical Problems
In the 9th C, Mahaviracharya ( Mysore) wrote Ganit Saar Sangraha where he described the currently used method of calculating the Least Common Multiple (LCM) of given numbers. He also derived formulae to calculate the area of an ellipse and a quadrilateral inscribed within a circle.In the late 9th C, Sridhara (probably Bengal) provided mathematical formulae for a variety of practical problems involving ratios, barter, simple interest, mixtures, purchase and sale, rates of travel, wages, and filling of cisterns. Some of these examples involved fairly complicated solutions and his Patiganita is considered an advanced mathematical work.The study of mathematics appears to slow down after the onslaught of the Islamic invasions and the conversion of colleges and universities to madrasahs. But this was also the time when Indian mathematical texts were increasingly being translated into Arabic and Persian. Although it appears that original work in mathematics ceased in much of Northern India after the Islamic conquests, Benaras survived as a center for mathematical study, and an important school of mathematics blossomed in Kerala. Madhava (14th C, Kochi) made important mathematical discoveries that would not be identified by European mathematicians till at least two centuries later. His series expansion of the cos and sine functions anticipated Newton by almost three centuries. Important discoveries by the Kerala mathematicians included the Newton-Gauss interpolation formula, the formula for the sum of an infinite series, and a series notation for pi.Why, one might ask, did Europe take over thousand years to attain the level of abstract mathematics achieved by Indians such as Aaryabhatta? The answer appears to be that Europeans were trapped in the relatively simplistic and concrete geometrical mathematics developed by the Greeks. It was not until they had, via the Arabs, received, assimilated and accepted the place-value system of enumeration developed in India that they were able to free their minds from the concrete and develop more abstract systems of thought. This development thus triggered the scientific and information technology revolutions which swept Europe and, later, the world. The role played by India in the development is no mere footnote, easily and inconsequentially swept under the rug of Eurocentric bias. To do so is to distort history, and to deny India one of it's greatest contributions to world civilization.
By:
Arjun Pal
Knowledge Cell - Globsyn Business School
