सर्वदोर्युतिदलं चतुःस्थितं बाहुभिर्विरहितं च तद्धतेः ।मूलमस्फुटफलं चतुर्भुजे स्पष्टमेवमुदितं त्रिबाहुके ॥Take the sum of all sides, and divide it into half. Multiply the result by the four differences of itself with each side. The square root of the result is is the approximate area of a quadrilateral, and precise for a triangle. (This is exact for …
Monthly Archives: October 2021
Proving सर्वदोर्युतिदलं for cyclic quadrilaterals
With the jyā recurrence without the standard radius having been demonstrated in the previous article, we return to cyclic quadrilaterals, and the proof of Bhāskarācārya’s verse on the areas of cyclic quadrilaterals and triangles: सर्वदोर्युतिदलं चतुःस्थितं बाहुभिर्विरहितं च तद्धतेः ।मूलमस्फुटफलं चतुर्भुजे स्पष्टमेवमुदितं त्रिबाहुके ॥Take the sum of all sides, and divide it into half. Multiply …
Continue reading “Proving सर्वदोर्युतिदलं for cyclic quadrilaterals”
Finding Jyā Recurrence Without The Radius – Part 2
Using the sampurṇajyā relationship we saw in the previous article, we can now achieve what we set out to – find a jyā recurrence without using the standard radius. First, recollect the sampurṇajyā relationship. Since , denoting the पठितज्याः (tabular jyā) as as usual, we can say: Thus, knowing the first two tabular jyā, we …
Continue reading “Finding Jyā Recurrence Without The Radius – Part 2”
Diagonals and Area of Cyclic Quadrilaterals
In the previous article in this series, we began a search for a jyā recurrence without using the standard radius R, and went off into triangles and cyclic quadrilaterals. In this article, we will dive deeper into the latter. The sampurṇajyā relationship First, we consider an relationship about sampurṇajyā and associated arcs (चाप) in a …
Continue reading “Diagonals and Area of Cyclic Quadrilaterals”
Finding Jyā Recurrence Without The Radius – Part 1
The Kerala School, as we saw in past articles, were big on options for jyā computations. First, they improved Aryabhata’s jyā recurrence, then added on an improved interpolation method. Later, we saw the Mādhava jyā series, plus a partially pre-computed version of it. Finally, we have the श्रेष्ठं नाम वरिष्ठानां verse, which provides a precomputed …
Continue reading “Finding Jyā Recurrence Without The Radius – Part 1”
