The Calculation of Sines

Two lines should be drawn. These lines, which take the form of the Full Chord: sampūrṇa-jīvā corresponding to twice the degrees of the Complementary Arc: koṭyaṃśa of the smaller and larger arcs, are effectively the same as the Cosine: koṭijyā (the sine of the complementary arc) in the standard Radius: trijyā circle. Similarly, a single line should be drawn to touch the two endpoints of those lines. This line—even though it represents the Full Chord of twice the sum of the two arcs—is actually the Sine of the Sum of Arcs: cāpaikya-jyā within the radius-circle. Thus, considering the Sines and Cosines of both the small and large arcs—

The diagram shows a circle containing an irregular quadrilateral. A horizontal line, labeled "Radius-diameter" (trijyā-vyāsa), serves as a diagonal. Other lines represent the "Sine of the Sum" (yoga-jīvā), the "Sine of the Larger Arc" (vṛddha-jyā), and the "Sine of the Smaller Arc" (laghu-jyā). Various triangles are formed within the circle to facilitate trigonometric proofs using the properties of cyclic quadrilaterals.

—an irregular quadrilateral figure is produced. One diagonal of this figure, which passes through the east and west markers, is visibly the Radius itself, because the circle was constructed using half the radius. The second diagonal, which measures the Sine of the sum of the arcs, is not immediately visible but is implied by the geometry.

In this context, by applying the mathematical rules from the traditional texts—starting from the verse "The desired diagonal should first be assumed..." and ending with "the square root of the sum of the squares of the perpendiculars"—one can easily understand the second diagonal, which represents the Sine of the sum of the arcs. Kamalākara is referring to standard rules for solving quadrilaterals, likely from the Līlāvatī or Siddhānta Śiromaṇi of Bhāskara II. These rules allow a mathematician to find the diagonal of a cyclic quadrilateral if the sides are known. This is achieved by calculating the Perpendiculars: lamba and the Base-segments: ābādhā of the two triangles formed on either side of the known Radius-diagonal.

Specifically, in a triangle, by using the rule "the sum of the two sides," the segment related to the small arc...