Origin of Hindu Geometry
The Hindu Geometry originated in a ·very remote age in connection with the construction of the altars for the Vedic sacrifices. The sacrifices, as described in the Vedic literature of the Hindus, were of various kinds. The performance of some of them was obligatory upon every Vedic Hindu, and hence they were known as Nitya (or “obligatory”, “indispensable”). Other sacrifices were to be performed each with the purpose of achieving some special object.
Those who did not aim at the attainment of any such object had no need to perform any of them. These sacrifices were classed as Kamya (or ”optional”, ”intentional”). According to the strict injunctions of the Hindu scriptures, each sacrifice must be made in an a] tar of prescribed shape and size. It was emphasized that even a slight irregularity and variation in the form and size of the altar would nullify the object of the whole ritual and might even lead to an adverse effect. So the greatest care had to be taken to secure the right shape and size of the altar.
In this way there arose in ancient India problems of geometry and also of arithmetic and algebra. There were multitudes of altars. Let us take for instance the three primary ones, viz. the Garhapatya, Aha-,aniya and Dak~l1Ja, in which every Vedic Hindu had to offer sacrifices daily.
The Ga,.hapatya altar was prescribed to be of the form of a square, according to one school, and of a circle according to another. The Aha-,aniya altar had always to be square and the Dak~i’l}a altar semi-circular. But the area of each had to be the same and equal to one square -,yama 1 • So the construction of these three altars involved three geometrical operations :
(i) to construct asquare on a given straight line ;
(ii) to circle a square and vice versa ; and
(iii) to double a circle. The last problem is the same as the evaluation of the surd ,[2. Or it may be considered as a case of doubling a square and then circling it. There were altars of the shape of a falcon with straight or bent wings, of a square, an equilateral triangle, an isosceles trapezium, a circle, a wheel (with or without spokes), a tortoise, a trough and of other complex forms all having the same area. Again at the second and each subsequent construction of an altar, it was necessary to increase its size by a specified amount, usually one square puru~a. 11 but the shape was always kept similar to that of the first construction. Thus there arose problems of equivalent areas and transformation of areas. The Vedic geometers also treated problems of ‘application of areas’.