Jya-Shreni: Sine and Cosine Series Before Taylor

How Madhava of Sangamagrama wrote down the power series for sine and cosine in Kerala around 1400, three centuries before Brook Taylor

Around 1400 CE, Madhava of Sangamagrama wrote down the infinite power series for sine and cosine as a sequence of corrective refinements to the angle itself. The series are what modern mathematics calls the Taylor expansions. Madhava had them three centuries before Taylor, derived them rigorously, and used them to build an 8-decimal-accurate sine table.

The Sine In Your Phone

Open the calculator on your phone. Type the sine of thirty-seven degrees. The screen shows 0.6018150232. That number did not come from a lookup table. The phone's math library computed it, in a few microseconds, by evaluating a polynomial. That polynomial is the truncation of an infinite series for sine. And the earliest known author of that series was not Brook Taylor in 1715, nor Isaac Newton in 1669. It was a Nambudiri Brahmin named Mādhava of Saṅgamagrāma, working with an iron stylus on a palm-leaf folio in a Kerala village around 1400 CE, roughly three hundred years earlier. The series Mādhava scored onto the leaf, in modern notation, looks like this.

sin x = x − x³/3! + x⁵/5! − x⁷/7! + ...

And the companion for cosine.

cos x = 1 − x²/2! + x⁴/4! − x⁶/6! + ...

Every calculus student who has ever opened a textbook has seen those two lines. They are almost always attributed to Brook Taylor, who published his general expansion theorem in 1715, or to Isaac Newton, who had the sine series by 1669. The attribution is historically wrong by about two and a half centuries. The person who first wrote those series down, in verse, in Sanskrit, in southern India, was a Nambudiri Brahmin named Mādhava of Saṅgamagrāma, working around 1400 CE.

Who Mādhava Was

Mādhava (c. 1340 to 1425 CE) lived in Saṅgamagrāma, a village near modern Irinjalakuda in central Kerala, about sixty kilometres from the port of Cochin. No original manuscript of his own survives. What we have of him is quotations preserved by his students and his students' students, a lineage that ran for the next two centuries through Parameśvara, Nīlakaṇṭha Somayājī, Jyeṣṭhadeva, Śaṅkara Vāriyar, and Acyuta Piṣāratī. Each of them names Mādhava as the source of the infinite series results. The Tantrasaṅgraha of Nīlakaṇṭha, composed in 1501, explicitly attributes the sine and cosine series to Mādhava. The Yuktibhāṣā of Jyeṣṭhadeva, composed around 1530 in Malayalam, preserves the full derivation and credits Mādhava at every step.