Bhāskara’s Approximation to and Mādhava’s Series for Sine: A Mini-Primary Source Project for Calculus 2 Students

While chords on circles were studied extensively in ancient Greek geometry (for example, Euclid’s Elements contains plenty of theorems relating the circle’s arc to the subtended line segment), it was the mathematicians and astronomers of India who first calculated values of half-chords instead [Gupta 1967, 121]. This work led very directly to the function that we call sine today. The Primary Source Project Bhāskara's Approximation to and Mādhava's Series for Sine uses mathematical results developed in medieval India as a means to enrich students' understanding of the process of approximating a transcendental function (e.g., sine) by an algebraic one (e.g., rational, polynomial).

Rendering of Bhāskara's original Sanskrit description of his sine approximation [Gupta 1967, p. 122].

An enormous amount of work in Indian mathematics was dedicated to sine calculations, motivated primarily by astronomy. One of the stunning results obtained from those efforts is the incredibly accurate seventh-century approximation for sine given by Bhāskara I (c. 600–c. 680) in his first work, now called Mahābhāskarīya. Translated into modern notation, this becomes the very mysterious approximation

\[\sin(x)≈\frac{16x(\pi−x)}{5\pi^2−4x(π−x)} \,\,\, \mbox{ for } x\in[0,\pi].\]

Bhāskara's work built off of earlier knowledge found in the Āryabhaṭīya, the only surviving work of the fifth-century Indian mathematician Āryabhaṭa (476–550). Much later, Mādhava of Saṇgamagrāma (c. 1350–c. 1425) constructed an infinite series expansion for sine, which is equivalent to the standard power series formula still taught in calculus courses today.

First satellite built and launched by India, named after Āryabhaṭa in honor of the astronomical impact of his work.
Photo credit: NASA, via Indian Space Research Organisation.

This project guides students through an analysis of short excerpts from the three above-mentioned mathematicians’ work. Its focus is not primarily on how they originally came up with these results, as their methods are mostly unknown. Additionally, the best guesses that historians of mathematics have proposed with regard to how these results were developed involve far more geometry than one typically includes in a second-semester Calculus course. (For one such method, see [Van Brummelen 2009, 104].) Accordingly, the project's focus is instead on comparing and contrasting the methods to each other, as well as to the modern power series treatment for sine that is typically presented in a Calculus 2 course. The student should leave the project with an understanding of other frameworks for algebraically approximating transcendental functions, as well as some of the many contributions of Indian astronomers to modern mathematics. Along the way, the project provides a substantial amount of practice with standard second-semester Calculus competencies such as finding power series and applying Taylor’s Error Theorem.

The complete project Bhāskara's Approximation to and Mādhava's Series for Sine (pdf) is ready for student use, and the LaTeX source code is available from the author by request.  A set of instructor notes that explain the purpose of the project and guide the instructor through the goals of each of the individual sections is appended at the end of the student project.

This project is the nineteenth in A Series of Mini-projects from TRIUMPHS: TRansforming Instruction in Undergraduate Mathematics via Primary Historical Sources appearing in Convergence, for use in courses ranging from first year calculus to analysis, number theory to topology, and more. Links to other mini-PSPs in this series appear below. The full TRIUMPHS collection includes eleven additional mini-PSPs for use in teaching standard topics from the first-year calculus curriculum. The Convergence series Teaching and Learning the Trigonometric Functions through Their Origins by Daniel E. Otero also offers several mini-Primary Source Projects designed to serve students as an introduction to the study of trigonometry, including a project based on another Indian mathematical work: Varāhamihira and the Poetry of Sines.

Acknowledgments

The development of the student project Bhāskara's Approximation to and Mādhava's Series for Sine has been partially supported by the TRansforming Instruction in Undergraduate Mathematics via Primary Historical Sources (TRIUMPHS) project with funding from the National Science Foundation’s Improving Undergraduate STEM Education Program under Grants No. 1523494, 1523561, 1523747, 1523753, 1523898, 1524065, and 1524098. Any opinions, findings, and conclusions or recommendations expressed in this project are those of the author and do not necessarily reflect the views of the National Science Foundation.

References

Gupta, Radha Charan. 1967. Bhāskara I’s Approximation to Sine. Indian Journal of History of Science 2(2):121–136.

Van Brummelen, Glen. 2009. The Mathematics of the Heavens and the Earth: The Early History of Trigonometry. Princeton NJ: Princeton University Press. ISBN: 978-0-691-12973.