The Science of the Shulba: A Study in Early Hindu Geometry c9.20

c9.20

The description of the division of a rectangle or a square by diagonals is found in the *Sulba* primarily in connexion with the making of bricks of desired size and shape. Certain interesting geometrical theorems have been assumed there. The brick which resembles in shape the portion of a rectangle or of a square divided by its diagonal is termed *ardhyā* (or "the half") and that resembling a portion by the two diagonals is called *pādyā* (or "the quarter"). There are distinguished two kinds of *pādyā* of a rectangle, *viz.*, *dīrgha-pādyā* (or "longish or broader quarter") and *śūla-pādyā* (or "trident quarter").¹ This distinctive nomenclature implies: (1) the halves of a rectangle or of a square by a diagonal are identical in size as well as in shape, and (2) so are also the quarters of a square by its diagonals; and (3) the diagonals of a rectangle divide it into four parts which are equal in area but they are of two kinds as regards their shape. These names perhaps further imply an idea of *obtuse* and *acute* angles. There are bricks which are halves of the quarter bricks by the perpendicular from the vertex on the base. No distinction is found to have been made between the half of a *dīrgha-pādyā* and that of a *śūla-pādyā*, which clearly shows that the Śulba-kāras were aware that those halves were identical. Thus it appears that the early Hindu geometers knew the simple cases of the congruence theorems.