Aryabhata and the Construction of the First Trigonometric Table

TL;DR

This paper explores Aryabhata's pioneering method for constructing the first sine table, highlighting his use of difference formulas and their connection to finite difference calculus, with pedagogical insights and exercises.

Contribution

It reveals Aryabhata's original approach to generating sine tables using difference formulas, linking ancient methods to modern calculus concepts.

Findings

Aryabhata's sine difference formula predates modern calculus.

His method involves similar triangles and difference identities.

The paper provides pedagogical exercises for understanding these concepts.

Abstract

Few among us would know that the first mention of the sine and the enumeration of the first sine table are to be credited to Aryabhata. The method to generate this relies on the sine difference formula which is derived using ingenious arguments based on similar triangles. We describe how this was done. In order to make our presentation pedagogical we take a unit circle and radians instead of the (now) archaic notation in the Aryabhatiya and its commentators. We suggest a couple of exercise problems and invite the enterprising student to try their hands. We also point out that his sine and the second sine difference identities are related to the finite difference calculus we now routinely use to calculate derivatives and second derivatives. An understanding of these trigonometric identities and preparation of the sine table will enable a student to get an appreciation of the path breaking work of Aryabhata.