The solution of a first-degree indeterminate equation by a method called kuttaka (or "pulveriser") was given by Aryabhata I. This method of finding integer solutions resembles the continued fraction process and can also be seen as a use of the Euclidean algorithm.
Brahmagupta seems to have used a method involving continued fractions to find integer solutions of an indeterminate equation of the type ax + c = by. Prthudakasvami's commentary on Brahmagupta's work is helpful in showing how "algebra", that is the method of calculating with the unknown, was developing in India. Prthudakasvami discussed the kuttaka method which he renamed as "bijagnita" which means the method of calculating with unknown elements.
To see just how this new idea of algebra was developing in India, we look at the notation which was being used by Prthudakasvami in his commentary on Brahmagupta's Brahma Sputa Siddhanta Ⓣ. In this commentary Prthudakasvami writes the equation 10x + 8 = x2 + 1 as:
yava 0 ya 10 ru 8Here yava is an abbreviation for yavat avad varga which means the "square of the unknown quantity", ya is an abbreviation for yavat havat which means the "unknown quantity", and ru is an abbreviation for rupa which means "constant term". Hence the top row reads
yava 1 ya 0 ru 1
0x2 + 10x + 8while the second row reads
x2 + 0x + 1The whole equation is therefore
0x2 + 10x + 8 = x2 + 0x + 1or
10x + 8 = x2 + 1.
Article by: J J O'Connor and E F Robertson