The Science of the Shulba: A Study in Early Hindu Geometry c15.64

c15.64

This supposition is doubtless wrong. For in the first expression the word *jñeyāni* very clearly qualifies *vedi-viharaṇāni* or "the methods of construction of the *vedi*," the word *vedi* undoubtedly referring to the *Mahāvedi* mentioned in the foregoing rules. This is further confirmed by the word *etāvanti* meaning "these," "so many," the reference being to the four methods of constructing the *Mahāvedi* described by Āpastamba. Where is then the ground to suppose, as has been done by Thibaut and Bürk, that *jñeyāni* implies the rectangles employed for the purpose of those methods? In the second passage the word *viharaṇam* ("construction" or "the method of constructions") clearly ¹ refers to the methods of construction described in the two foregoing rules, *viz*., *ĀpŚl*, i. 2 and 3, and the word *jñeyābhiḥ* to the rational rectangles used therein. This has been admitted by Bürk also. Now the methods referred to are those for the construction of squares and rectangles of given sides, and they primarily depend upon drawing right-angled triangles (or rectangles) having a given side, or more particularly, having a side equal to a side of the required figure. The sides and the diagonal of the rectangles are stated to be (*a*, 5*a*/12, 13*a*/12) and (*a*, 3*a*/4, 5*a*/4). It is only when *a* is a multiple of 12 (in the first case) or a multiple of 4 (in the second case) that the sides and the diagonal of the rectangles employed for the purpose of the construction of altars, will be expressible in rational integers, otherwise not. Indeed in actual practice Āpastamba has employed four particular cases of the first, *viz.*, (2½, 6, 6½), (2⅟₁₂, 5, 5⅟₁₂), (4⅙, 10, 10⅙) and (11¼, 27, 29¼), which have fractional sides and diagonals.¹