Yuktibhāṣā does not explicitly calculate the volumes of pyramids or cones, but it would be instructive to use the method of Sankalita to do so, to demonstrate the generalizability of this method beyond what is explicitly shown. Volume of a Pyramid Consider a square pyramid of side a, and height h. We divide it into …
Category Archives: Indian Mathematics
Concepts of aṇu and parārdha
Since the 19th century, modern calculus has relied on ideas of limits for its theoretical grounding. The infinitesimal, favoured by Leibniz, was used to describe and derive results in calculus prior to that. aṇu and parārdha Kerala Calculus relies on two concepts, aṇu and parārdha, to ground and derive their results. For example, the मूलसङ्कलितम् …
Generalizing the Calculus of the Kerala School
What we have seen so far of the Kerala School is quite interesting, to say the least. We have seen: A line integral (circumference of a circle). Areas of a circle and sphere and Volume of a sphere. Integrals (sankalitas) for powers. Power series approximation for what is called today. Trigonometry similar to the modern …
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Proving सर्वदोर्युतिदलं for triangles
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 …
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 …
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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 …
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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 …
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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 …
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Surface Area of a Sphere
In a previous article, we saw how the area of a circle, and the volume of a sphere are computed using Kerala School methods. We had not looked at the surface area of a sphere at that point, since we had not looked at jyā yet. Now that that is familiar to us, it is …
जीवे परस्परन्यायः – The sin(A+B) rule and some trigonometric reasoning
Alongside calculus, the Kerala School also came up with some interesting results in Trigonometry, including the now commonly known method to calculate the sines (ज्या) of the sums are difference of two angles (arcs): जीवे परस्परनिजेतरमौर्विकाभ्याम् अभ्यस्य विस्तृतिदलेन विभज्यमाने ।।अन्योन्ययोगविरहानुगुणे भवेतांयद्वास्वलम्बकृतिभेदपदीकृते द्वे ।। 1) The two jyā (of the two arcs), multiplied by the kojyā …
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