Previously, we saw the nihatya series for jyā and śara and how that enables a very accurate estimate of those functions for any arc, accurate to the fourths. We also saw how this led to the partially precomputed vidvān series as well as the table of पठितज्या accurate to thirds – श्रेष्ठं नाम वरिष्ठानाम् etc. …
Category Archives: Indian Mathematics
A Better Bowstring: The Mādhava Jyā Series
What’s better than an improved jyā table and an improved method for interpolation? A full jyā series that lets you compute jyā of any arc, to any degree of precision you wish. निहत्य चापवर्गेण चापं तत्तत्फलानि च । हरेत् समूलयुग्वर्गैस्त्रिज्यावर्गाहतैः क्रमात् ॥चापं फलानि चाधोऽधो न्यस्योपर्युपरित्यजेत् । जीवापत्यै संग्रहोऽस्यैव विद्वान् इत्यदिना कृतः ॥ Multiply the arc by …
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Fun with Sankalitas – Part 2
Previously, we saw the मूलसङ्कलितम् and also noted that as n becomes large, . We also saw the समघातसङ्कलितानि, sums of higher powers, including the sum of squares – वर्गसङ्कलितम्, and sum of cubes – घनसङ्कलितम्. We noted that in the limit of large n, the kth power sankalita tends to: We also saw सङ्कलितसङ्कलितम्, …
Improving the jyā table
Recap In the previous article in this series, we saw how Āryabhaṭa’s jyā table was calculated. It was a remarkably simple recurrence relation, with accuracy to one minute of arc (using the Indian standard circle of circumference 21600′) at 24 points 225′ apart. प्रथमाच्चापज्यार्धाद्यैरूनं खण्डितं द्वितीयार्धम् ।तत्प्रथमज्यार्धांशैस्तैस्तैरूनानि शेषाणि ।। We set both first jyā B1 …
Stringing the bow – a backgrounder on Indian trigonometry and the concept of jyā
Having calculated the ratio of the circumference and diameter of a circle to any accuracy we want with the Madhava series, we move onto the question of calculating ज्या/jyā or the half-chord of any arc. ज्या is the Indian counterpart of the modern sine function, and a backgrounder on how Indians thought of it, and …
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Calculating Pi to eleven digits by hand using the Madhava Series – चण्डांशुचन्द्राधमकुम्भिपालः
31415926536 A famous verse in a later work from the Kerala school, with a कटपयादि mnemonic of pi to ten decimal places goes: आनूननून्नाननुनुन्ननित्यैस् समाहताश्चक्रकलाविभक्ताः । चण्डांशुचन्द्राधमकुम्भिपालैर्व्यासस्तदर्धं त्रिभमौर्विका स्यात् ।। करणपद्धतिः – पुतुमनसोमयाजी The circumference of a circle in minutes of arc is multiplied by आनूननून्नाननुनुन्ननित्यम् (=10000000000) and divided by चण्डांशुचन्द्राधमकुम्भिपालः = (31415926536) to yield the diameter …
Making the Madhava series practically useful
The Madhava series is slow and boring, says … Madhava यत्सङ्ख्ययात्र हरणे कृते निवृत्ता हृतिस्तु जामितयातस्या ऊर्ध्वगता या समसङ्ख्या तद्दलं गुणोऽन्ते स्यात् ||तद्वर्गो रूपयुतो हारो व्यासाब्धिघातात् प्राग्वत् ताभ्यामाप्तं स्वमृणे कृते धने क्षेप एव करणीयः ||लभ्धः परिधिः सूक्ष्मो बहुकृत्वो हरणतोऽतिसूक्ष्मश्च ||When you stop division out of boredom, remember the last divisortake the next even number, and halve …
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cāpı̄karaṇam: The arctangent function
After our detour to take a detailed look at sankalitas, we go back to the व्यासे वारिधिनिहते series and – following the reasoning in Yuktibhāṣā – see how we can modify it to convert any ज्या / कोटिः pair to its चापः (arc). In modern terms, this corresponds to finding an infinite series for the …
Area of a circle, and volume of a sphere
In the previous article in this series, we saw how sankalitas are the Kerala calculus analogue to integrals in modern calculus, and how they were computed in the limit of large n. In this article, we can take a look at how these were used to compute the volume of a sphere. Area of a …
Fun with Sankalitas
In the previous article in the series, we saw how the famous Madhava circumference (Pi) series was justified using an argument based on the geometry of Kerala roofs, and the mathematical techniques of shodyaphala and sankalita (सङकलितम्). Since the idea of sankalita is central to the Calculus of the Kerala school, it makes sense to …
