The mensuration of quadrilaterals and the generation of Pythagorean triads: A mathematical,heuristical and historical study with special reference to Brahmagupta's rules

Summary The ancient rule for the area of a quadrilateral,, is examined to show its inaccuracy and its arbitrariness, and in order to see how those using it might have become aware of its shortcomings. Consideration ofBrahmagupta's vastly more sophisticated rule,, and of the Hindu schedule of recognized quadrilateral types, leads to the question of the proper assessment to be made of this strand of Indian mathematics. The work on quadrilaterals was intimately connected with the generation of Pythagorean triangles out ofbja (=seed) numbers by the same method as had probably been used by the mathematicians of Old Babylonia. Ways in which this procedure could have arisen by induction from particular numerical calculation are shown and the rest of the study consists primarily of an investigation as to whether the same kind of approach might not also have been used to obtain the remarkable rules on quadrilaterals given byBrahmagupta. It is found that there is no compelling need to adopt the common assumption thatBrahmagupta must have had access to Alexandrian mathematics. But even if he did chance to learn of some helpful items from the works ofHero orPtolemy, it still seems necessary for a proper appreciation of his understanding of the mensuration of quadrilaterals to suppose that he would have worked such items into the contemporary Indian mathematical context in some such way as is indicated here. In particular, it is argued thatBrahmagupta, far from indulging in reckless generalization, could well have proceeded with great caution, from one amenable type of quadrilateral to another, verifying his inductions by comparing them with results given by alternative calculation methods.