Teaching and Learning the Trigonometric Functions through Their Origins: Episode 4 – Varāhamihira and the Poetry of Sines

Daniel E. Otero (Xavier University)

Until the twenty-first century, one typically learned about matters of general knowledge from printed books, such as the great Encyclopedia Britannica, that premier source of general information in the English language. Britannica was first published in 1771, and the second version of its fifteenth edition came out in 1985, a 30-volume series weighing a total of more than 125 pounds and indexing over 500,000 topics! This was to be its final printed edition: in 2003, its full contents were released on a single CD-ROM, a plastic disk 5 inches in diameter and weighing a bit more than half an ounce![1]

Encyclopedia Britannica, fifteenth ed., 1985, thirty print volumes
Encyclopedia Britannica, fifteenth edition, 1985.

Encyclopedia Britannica, 2003.
Encyclopedia Britannica, 2003.

Now suppose you were faced with a large volume of mathematical information that you need to store and to pass along to users who come after you, but you do not know how to write. In an oral culture before the widespread use of writing, data storage problems like the one we have described here were often handled through the mechanics of poetry: metrical patterns of verse allowed the natural skills of human users to commit great amounts of information to memory for reliable transmission over time. The classroom unit, Varāhamihira and the Poetry of Sines, presented here introduces just such a process.

In the sixth century, the Hindu astronomer Varahāmihira (505–587 CE) composed a treatise called Pañcasiddhāntikā (The Five Canons), a summary of the Greek, Egyptian, Roman and Indian astronomy of his day. This astronomy carried on the Greek tradition that applied the geometry of the circle and sphere to work out the positions of the stars and planets in the heavens, a tradition that in its Greek past used to employ tables of chords in circles associated with the arcs they spanned, but that in Hindu science shifted to tables of sines of these arcs. To preserve the information in such a table over time, it was set to verse and recited. In just 11 lines, Varāhamihira encoded a full table of sines for 24 arcs from 0° to 90° (in multiples of 3°45´), as well as statements of a handful of trigonometric theorems.

Sanskrit verses from Varāhamihira’s Pañcasiddhāntikā
Sanskrit verses from Varāhamihira’s Pañcasiddhāntikā (IV, 9-10).

In the classroom unit, students examine the first 9 lines of Varāhamihira’s verse, in English translation, and use the information encoded in these lines to build a partial table of sines for angles up to 60°. An optional appendix provides (in verses 10–11 of the source text) the information students need to extend that table to arcs of measure 90°. Because the context of the source text is astronomical, the unit includes a treatment of some basic astronomy having to do with the motion of the Sun through the zodiacal constellations. Students are also provided with a brief introduction to how circular arcs (and, equivalently, the central angles they span) are measured, as well as a discussion of the trigonometric innovation by Hindu astronomers that replaced the chord with the sine as the trigonometric quantity of chief interest

This classroom unit is one of a series of six units in the Convergence series Teaching and Learning the Trigonometric Functions through Their Origins, which attempts to provide students with a rich motivation for the study of trigonometry by presenting a series of episodes from the long history of the subject. These episodes work to contextualize the main ideas of trigonometry in the questions that earlier mathematicians addressed in developing the subject over thousands of years.

The unit Varāhamihira and the Poetry of Sines (pdf) is ready for student use. It is meant to be completed in two 50-minute classroom periods, plus time in advance for students to do some initial reading and time afterwards for them to write up their solutions to the tasks. A brief set of instructor notes offering additional background and practical advice for the use of these materials in the classroom is appended at the end of the student version of these projects.

Varāhamihira and the Poetry of Sines is the fourth in the Convergence series Teaching and Learning the Trigonometric Functions through Their Origins. Although these classroom projects are posted here as parts of a series, each episode has been redesigned to stand alone. Readers who want to see the entire Primary Source Project (PSP) from which these units are drawn can obtain that PSP, A Genetic Context for Understanding the Trigonometric Functions, without waiting for future installments to appear, from the website of the NSF-funded project TRansforming Instruction in Undergraduate Mathematics via Primary Historical Sources (TRIUMPHS). The LaTeX source code of all TRIUMPHS projects, including the units to appear in this series, are available from the project authors by request.

[1] In fact, the 2000 launch of Wikipedia, the instantly editable online encyclopedia, has come to dominate the landscape of general-purpose reference works in English. Even optical disk technology for the written word has become obsolete with the migration of information storage to the “cloud.” In 2021, Wikipedia counted nearly 150,000 contributors worldwide responsible for more than 6 million articles on topics of all sorts.