Mathematics in the classical era
By Prof. Vijaya Kumar Murty
The most famous mathematicians of the classical era were Aryabhata I (500 CE), Brahmagupta (700 CE), Bhaskara I (900 CE), Mahavira (900 CE), Aryabhata II (1000 CE), and Bhaskaracharya or Bhaskara II (1200 CE).
During this period, two centers of mathematical research emerged, one at Kusumapura near Pataliputra and the other at Ujjain. Aryabhata I was the dominant figure at Kusumapura and may even have been the founder of the local school. His fundamental work, the Aryabhatiya, set the agenda for research in mathematics and astronomy in India for many centuries.
One of Aryabhata’s discoveries was a method for solving linear equations of the form ax + by = c. Here a, b, and c are whole numbers seeking values of x and y in whole numbers satisfying the above equation. For example, if a = 5, b = 2, and c = 8, then x = 8 and y = −16 is a solution. In fact, there are infinitely many solutions:
x = 8 − 2m
y = 5m − 16
where m is any whole number, as can easily be verified. Aryabhata devised a general method for solving such equations called the kuttaka (or pulverizer) method. He called it the pulverizer because it proceeded by a series of steps, each of which required the solution of a similar problem, but with smaller numbers. Thus, a, b, and c were ‘pulverized’ into smaller numbers.
The Euclidean algorithm, which occurs in the Elements of Euclid, gives a method to compute the greatest common divisor of two numbers by a sequence of reductions to smaller numbers. As far as I am aware, Euclid does not suggest that this method can be used to solve linear equations of the above sort. Today, it is known that if the algorithm in Euclid is applied in reverse order, then in fact it will yield Aryabhata’s method. Unfortunately, the mathematical literature still refers to this as the extended Euclidean algorithm, mainly out of ignorance of Aryabhata’s work.
It should be noted that Aryabhata studied the above linear equations because of his interest in astronomy. In modern times, these equations are of interest in computational number theory and are of fundamental importance in cryptography.
Amongst other important contributions of Aryabhata is his approximation of π to four decimal places (3.1416). Also of importance is Aryabhata’s work on trigonometry, including his tables of values of the sine function, as well as algebraic formulae for computing the sine of multiples of an angle.
The other major center of mathematical learning during this period was Ujjain, which was home to Varahamihira, Brahmagupta, and Bhaskaracharya. The text Brahma-sphuta-siddhanta by Brahmagupta, published in 628 CE, dealt with arithmetic involving zero and negative numbers.
much of Brahmagupta's work was motivated by problems that arose in astronomy. He gave the famous formula for a solution to the quadratic equation ax² + bx + c = 0, namely x = (−b + √(b² − 4ac))/2a
It is not clear whether Brahmagupta gave just this solution or both solutions to this equation. Brahmagupta also studied quadratic equations in two variables and sought solutions in whole numbers. Such equations were studied only much later in Europe.
This period closes with Bhaskaracharya (1200 CE). In his fundamental work on arithmetic (titled Lilavati) he refined the kuttaka method of Aryabhata and Brahmagupta. The Lilavati is impressive for its originality and diversity of topics. Until recently, it was a popularly held view that there was no original Indian mathematics before Bhaskaracharya. However, the above discussion shows that his work was the culmination of a series of distinguished mathematicians who came before him. Also, after Bhaskaracharya, there seems to have been a gap of two hundred years before the next recorded work. Perhaps this is another time period about which more research is needed.
Notes & References
- By comparison, the Greeks were using the weaker approximation 3.1429