The Science of the Shulba: A Study in Early Hindu Geometry c14.s1.4

c14.s1.4

Now the doubling of a square, *i.e.*, the geometrical construction of √2, is necessary for the construction of the one of the three primarily essential altars of the Vedic sacrifices, *viz.*, the *Dakṣiṇa*. It has been pointed out before that the existence of those three altars is older than the *Ṛgveda* (before 3000 B.C.). Hence the theorem of the square of the diagonal, particularly in its simplest form for the case of the square, is as old as that. In connexion with the construction of the *Rathacakra-citi*, as in the case of the most of the *Kāmya Agni*, one has first to draw a square equal to the primitive and standard Agni, the *Caturasra-śyena-cit*, whose area is 7½ square puruṣas and whose form (Fig. 1) consists partly of squares and partly of rectangles. It will be easily seen that the solution of this problem is not possible without the help of the theorem of the square of the diagonal in its general form.¹ For the *Praüga-citi*, one has again to double that square. Hence the theorem must be as old as the *Taittirīya* and other *Saṃhitā* (*c.* 3000 B.C.) where, it has been pointed out before, occurs the express mention of the construction of the *Kāmya Agni*. In the *Śatapatha Brāhmaṇa*, we are asked to construct a square fourteen times as great as another square of one puruṣa. Again it is required to divide the small square into seven parts and three of the parts will have then to be combined with the larger square to form a new square.² From that *Brāhmaṇa*, we come to know of some ancient authorities who used to approve of the construction of a series of *Agni* of the square shape, with areas 1½, 2½,...6½ square puruṣas.³ These have been incidentally mentioned by Baudhāyana⁴ and Āpastamba.⁵ But the *Śatapatha Brāhmaṇa* particularly forbids the construction of such altars. That is, however, quite immaterial for the object we have now in view. It is sufficient and important for us to know that such constructions which clearly depend upon the addition of squares were once in vogue, at any rate in some particular schools, before the time of the *Śatapatha Brāhmaṇa* (*c.* 2000 B.C.). In one of the cases noted above it is necessary to transform a rectangle into a square before it can be added to another. In the *Śrauta-sūtra*, we find copious instances of the geometrical construction of √2, √3, etc. Some of those instances can be clearly distinguished from all the previous ones inasmuch as in them the application of the theorem of the square of the diagonal has been very clearly mentioned.¹ In the *Baudhāyana Śrauta*, the converse theorem is used for the construction of the *Mahāvedi*.² Thus we learn that the theorem of the square of the diagonal really plays a very important part in the science of altar-construction from very early times and in some cases it is, in fact, indispensable.