On the computation of the arcsin function in the Kerala school of astronomy and mathematics

TL;DR

This paper explores historical methods used by the Kerala school for computing the arcsin function, introduces a new approach based on these methods, and uncovers an unexpected integer sequence related to their iterative techniques.

Contribution

It provides a detailed analysis of four historical methods for arcsin computation from the Kerala school and presents a novel method for circle circumference calculation based on these techniques.

Findings

Four historical methods for arcsin computation are analyzed.

A new method for circle circumference calculation is introduced.

An unexpected integer sequence appears in the iterative arcsin method.

Abstract

This paper examines how the mathematicians and astronomers of the Kerala school tackled the problem of computing the values of the arcsin function. Four different approaches are discussed all of which are found in Nilakantha Somayaji's (1444 - 1545 CE) Tantrasangraha and the roots of all of which can be traced to ideas originally articulated by Sangamagrama Madhava (c. 1340 - 1425 CE): (i) a simple method when the argument is small; (ii) an iterative method when the argument is small; (iii) a method based on a lookup table; (iv) a method when the argument is large. The paper also contains the original Sanskrit verses describing the various methods and English translations thereof. Moreover, there is a presentation of a novel method for computing the circumference of a circle found in Jyeshthadeva's (c. 1500 - 1575 CE) Yuktibhasha which is based on method (i) for computing the arcsin function. All methods have been illustrated with numerical examples. A surprising by-product of the investigation is a totally unexpected appearance of a core integer sequence, namely, the entry A001764 in the Online Encyclopedia of Integer Sequence, while studying the iterative method for computing the arcsin function.