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 …
Monthly Archives: April 2021
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 …
The logic behind व्यासे वारिधिनिहते …
व्यासे वारिधिनिहते रूपहृते व्याससागराभिहते । त्रिशरादिविषमसंख्याभक्तमृणं स्वं पृथक्क्रमात् कुर्यात् ॥from the diameter multiplied by four and divided by one reduce and add in turn, the diameter multiplied by four, and respectively divided by the odd numbers three, five and so on. In the previous article in this series, we made a brief acquaintance with the …
