ABSTRACT
From very early times in India, the emphasis on geometry has been from the numerical point of view and this has possibly been due to cultural reasons, since in India, most abstract concepts have had their origin in religion, or were intended for practical applications. The simplest geometrical figure, the right angled triangle(along with the so called theorem of Pythagoras) has already been mentioned in the Sulva Sutras2 (the earliest of which can be dated at least to the eighth century B.C.), was studied from the point of view of constructing Vedic altars. One of the commonest type of questions treated by the ancient Indians was the construction of right angled triangles all of whose sides are rational(integral), such as triangles with sides (3,4,5), (5,12,13), etc. which of course must have been probably known to other ancient civilisations too. The study of such triangles must
404 Math Unlimited
have continued to evolve in India and it is reasonable to believe that triangles with rational sides and rational areas were also studied here in India from very early times. However, the first documentary evidence of their study comes from the work of that great Indian astronomer-mathematician Brahmagupta(born 598 AD). He infact used such triangles for the construction of cyclic quadrilaterals with rational sides and rational diagonals. The aim of this article is to show how the study of such quadrilaterals which began in this “remote” part of the world, as far as the westerners were concerned, waited for thirteen centuries , before it came to be known in England due to the English translation of the work of Brahmagupta and Bhaskara II by (Colebrooke, 1817), crossed the Channel to move to France and eventually came to the attention of the nineteenth century German mathematician Kummer, who not only wrote a critical paper on this work of Brahmagupta, but also generalised the problem to construct(not necessarily cyclic) quadrilaterals with rational sides and diagonals by new methods, belonging to that part of mathematics now called ’Arithmetic Geometry’(which consists, in particular, of methods of solving indeterminate polynomial equations).
