Derivation of Sines

Abstract

In the manner explained above, derive the diameter for the circle of circumference measuring 21,600 minutes (cakra-kalā). Halve the diameter (to obtain the radius) and, with it, draw a circle. Draw through the centre (of the circle) the east-west and north-south lines and on both sides of the north-south line construct two equilateral triangles (sama-tryaśra). The sides of all these (four triangles) will be equal to the radius. Now, construct four complete chords (samasta-jyā) equal to the radius with their ends touching the ends of the north-south line. These will be the sides of the triangles. Then draw four radii starting from the centre and touching the tips of these four complete chords. Each of these will also be the sides (of the triangles drawn). Now, the halves of the north-south line will be the common sides of the triangles. Thus there will be two triangles on each side of the north-south line. Construct in this manner four equilateral triangles of sides equal to the radius. Here, construe that in each triangle, one side lies flat on the bottom which is called bhūmi (ground, base).